Probability and Statistics in Data Science using Python
7.3 An edX assignment has 50 multiple-choice questions, each with four choices of which one is correct. A student gets 3 points for solving a question correctly, and loses a point for an incorrect answer. What is the expected score of a student who answers all questions uniformly at random?
0
6.5 On any night, there is a 92% chance that an burglary attempt will trigger the alarm, and a 1% chance of a false alarm, namely that the alarm will go off when there is no burglary. The chance that a house will be burglarized on a given night is 1/1000. What is the chance of a burglary attempt if you wake up at night to the sound of your alarm?
0.084
5.6 Consider a die where the probability of rolling 1, 2, 3, 4, 5, and 6 are in the ration 1:2:3:4:5:6. What is the probability that when this die is rolled twice, the sum is 7?
0.12698
6.5 Suppose that 15% of the population have cancer, 50% of the population smokes, and 75% of those with cancer smoke. What fraction of smokers have cancer?
0.225
6.4 60% of our students are American, and 40% are foreign. 20% of Americans and 40% of foreigners speak two languages. What is the probability that a random student speaks two languages?
0.28
4.3 In how many ways can 7 distinct red balls and 5 distinct blue balls be placed in a row such that: 1. all red balls are adjascent 2. all blue balls are adjacent 3. no two blue balls are adjacent
1. 3628800 2. 4838400 3. 33868800
6.4 Let A and B be two random subsets of {1, 2,3,4}. What is the probability that A is a subset of B
0.3164
7.3 Each time you play a die rolling game you must pay $1. If you roll an even number, you win $2. If you roll an odd number, you lose additional $1. What is the expected value of your winnings?
0.50
For the uniform space {1, 2, ..., 10} find: 1. P({primes}) 2. P({multiples of 3})
1. 4/10 2. 3/10
3.4 (Cartesian Products) Does AxB and BxA have the same size for any sets A and B? 1. Yes 2. No
1
3.5 (Cartesian Powers) Let G = {0, 2, 4, 6, 8} what is |G^4|? 1. 5^4 2. 4^5 3. 5! 4. 0 + 2 + 4 + 6 + 8
1
4.1 0! = ? 1. 0 2. 1 3. infinite 4. undefined
1
5.7 Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations. Which is more probable? 1. Linda is a bank teller 2. Linda is a bank teller and is active in the feminist movement
1
2.4 Which of the following set pairs intersect? 1. {1, 2, 3} and {2, 4, 6} 2. {prime numbers} and {even numbers} 3. /emptyset/ and /emptyset/ 4. {/emptyset/, 1, 2} and {/emptyset/}
1 and 2
5.2 Which of the following outcomes are random (not certain) when rolling a six-sided die? 1. Get number 3 2. Get an even number 3. Get a positive number
1 and 2
6.2 Every event A is independent of: 1. emptyset 2. Omega 3. A itself 4. A^c
1 and 2
6.2 Roll two dice and let Fe be the event that the first die is even, S4 be the event that the second die is 4, and E0 the event that the sum of the first two dice is odd. Which of the following events are independent? 1. Fe and S4 2. Fe and E0 3. S4 and E0 4. Fe, S4, and Eo
1 and 2 and 3
6.2 Two dice are rolled. Let F3 be the event that the first die is 3, S4 the event that the second die is 4, and E7 the event that the sum is 7. Which of the following are independent: 1. F3 and S4 2. F3 and E7 3. S4 and E7 4. F3, S4, and E7
1 and 2 and 3
3.2 Which of the following are finite for every finite set A and infinite set B? 1. A intersect B 2. A union B 3. A - B 4. B - A 5. A symmetric difference B
1 and 3
3.4 (Cartesian Products) If A is finite and B is infinite then A x B can be: 1. empty 2. nonempty finite 3. infinite
1 and 3
7.2 Which 2 of the folllowing are true about the expectation of a random variable? 1. Not Random 2. Random Value 3. Property of the distribution 4. Independent of the distribution
1 and 3
6.2 Which of the following ensure that events A and B are independent 1. A and B^c are independent 2. A intersection B = emptyset 3. A is a subset of B 4. At least one of A or B is emptyset or Omega
1 and 4
6.5 A rare disease occurs randomly in one out of 10,000 people, and a test for the disease is accurate 99% of the time, both for those who have and don't have the disease. You take the test and the result is postive. The chances you actually have the disease are approximately:
1%
5.3 Which of the following implies P(S - T) = P(S) - P(T) for events S and T 1. T is a subset of S 2. T is a proper subset of S 3. S = T 4. S is a subset of T
1, 2, 3
5.3 Which of the following are events in the sample space: Omega = {1, 2, 3, 4, 5} 1. {1, 2, 3} 2. emptyset 3. Omega 4. {1} 5. {0, 3, 4}
1, 2, 3, 4
5.3 Which of the following holds for every event A? 1. P(A) >= 0 2. P(A) <= 1 3. P(A) + P(A^c) = 1 4. P(A) = P(A^c) 5. A = emptyset --> P(A) = 0 6. P(A) = 0 --> A = emptyset
1, 2, 3, 5
6.2 When rolling two dice, which of the following events are independent of the event that the first die is 4: 1. The second is 2, 2. The sum is 6, 3. The sum is 7 4. The sum is even
1, 2, 4
3.4 (Cartesian Products) Which of the following ensure that A x B = B x A? 1. A = B 2. A = emptyset 3. B = emptyset 4. A intersect B = emptyset 5. |A| = |B|
1, 2, and 3
3.1 Which of the following sets are finite? 1. Weeks in a year 2. Students at UCSD 3. Odd primes 4. Positive integer divisors of 30?
1, 2, and 4
4.6 Which of the following are equal? 1. 10 choose 4 2. 10 Choose 5 3 10 choose 6 4. 9 choose 5 + 9 choose 6
1, 3, 4
3.2 Recall that a square of an integer, for example, 1, 4 and 9 is called a perfect square. How many integers between 1 and 100 inclusive, are not perfect squares?
90
3.1 Which of the following sets are finite? 1. {X exists in Z | x^2 <= 10} 2. {X exists in Z | x^3 <= 10} 3. {X exists in N | x^3 <= 10} 4. {X exists in R | x^2 <= 10} 5. {X exists in R | x^3 = 10}
1, 3, 5. Explanation. 1. {-3, -2, ..., 3} 2. {x exists in Z| x <= 2} 3. {0, 1, 2} 4. False, not writing this shit out. 5. True {10.333333}
2.2 Which of the following are true? 1. 0 exists in {Even Numbers} 2. 0.5 exists in N 3. /emptyset/ exists in Q
1.
2.4 Which of the following holds 1. {3, 4} is not a proper superset of {3, 4} 2. {3, 4} != {3, 4} 3. {4, 3} is a proper subset of {3, 4} 4. {3, 4} is a proper subset of {4, 3}
1.
2.6 Which of the following statements hold for all A? 1. A x /emptyset/ = /emptyset/ 2. A x /emptyset/ = A 3. A is a subset of A^2 4. A exists in A^2 4. A x A^C = /emptyset/
1.
3.2 Which of the following pairs A and B satisfy |A union B| = |A| + |B|? 1. {1, 2} and {0, 5} 2. {1, 2} and {2, 3} 3. {English words starting with the letter 'a'} and {English words ending with the letter 'a'}
1.
4.2 Which of the following is larger for k <= n? 1. The number of k-permutations of an n-set 2. The number of k-subsets of an n-set
1.
4.3 In how many ways can a basketball coach select 5 starting players form a team of 15? 1. 15! / 5! * 10! 2. 15!/10! 3. 15!/5!
1.
5.3 Which of the following always hold for events A and B? 1. A is a subset of B --> P(A) less than or equal to P(B) 2. P(A) <= P(B) --> A is a subset of B
1.
5.4 Which of the following are independent repetitive experiments? 1. card draws with replacement. 2. card draws without replacement. 3. Neither
1.
5.5 What is the probability that two cards drawn from the same deck without replacement have the same rank? 1. 1/13 2. 1/17 3. 2/52
1.
6.2 Two dice are rolled. The event that the first die is 1 and the event that two dice sum up to b 7 are: 1. independent 2. Dependent
1.
6.3 The equality P(A intersection B = P(A) * P(B) holds whenever the events A and B are: 1. independent 2. disjoint 3. intersecting.
1.
7.1 Which of the following statements is correct? 1. Random variables are mappings between outcomes and real numbers 2. Random variables are mappsings between events and real numbers 3. Neither
1.
4.6 1. What is the coefficient of x^4 in the expansion of (2x - 1)^7 2. What is the constant term in the expansion of (x - 2/x)^6
1. -560 2. -160
2.1 Recall that /zero/ is the empty set. How many elements do the following sets have? 1. /zero/ 2. {/zero/} 3. {/zero/, /zero/} 4. {{/zero/}, /zero/}
1. 0 2. 1 3. 1 4. 2
6.1 Three fair coins are sequentially tossed. FInd the probability that all are heads if: 1. The first is tails. 2. The first is heads 3. At least one is heads
1. 0 2. 1/4 3. 1/7
5.5 Give cards are dealt from a poker deck. What is the probability of: 1. three-of-a-kind 2. two pairs 3. one pair
1. 0.0211 2. 0.0475 3. 0.4226
6.5 A car manufacturer has three factories producing 21%, 35%, and 44% of its cars, respectively. Of these cars, 7%, 6%, and 2%, respectively, are defective. A car is chosen at random from the manufacturer's supply. 1. What is the probability that the car is defective 2. Given that the car is defective, what is the probability that was produced by the first factory?
1. 0.0445 2. 0.3303
5.5 An instructor assigns 10 problems and says that the final exam will consist of 5 of them at random. If a student knows how to solve 7 of the problems, what is the probability that he or she will answer correctly: 1. all 5 problems 2. at least 4 problems
1. 0.083333333 2. 0.5
5.5 A 52 card deck is randomly split into 4 13 card hands. Find the probability that: 1. Each hand has an ace 2. One hand has all four aces.
1. 0.1055 2. 0.0106
5.6 Let P be a probability function on S= {a1, a2, a3}. Find P(a1) if: 1. P({a2, a3}) = 3P(a1) 2. P(a1) = 2P(a2) = 3P(a3)
1. 0.25 2. 0.5454
6.5 A car manufacturer receives its air conditioning units from 3 suppliers. 20% of the unitws come from supplier A, 30% from supplier B, and 50% from supplier C. 10% of the units from supplier A are defective, 8% of units from supplier B are defective, and 5% of units from supplier C are defective. If a unit is selected at random and is found to be defective. What is the probability that a unit came from supplier A if it is: 1. defective 2. non-defective
1. 0.289 2. 0.193
Suppose A, B are events that P(A) = 0.65, P(B) = 0.5 and P(A intersection B) = 0.25. What are the following probabilities? 1. P(A^c) 2. P(B^c) 3. P(A union B) 4. P(A - B) 5. P(B - 1) 6. P(A symmetric difference B) 7. P((A union B)^c)
1. 0.35 2. 0.5 3. 0.9 4. 0.4 5. 0.25 6. 0.65 7. 0.1
7.3 A bag contains five balls numbered 1 to 5. Randomly draw two balls from the bag and let X denote the sum of the numbers. 1. What is P(X <= 5)? 2. What is E(X)?
1. 0.4 2. 6
5.5 Roll three dice. What is the probability that the three outcomes: 1. contain at least a '1'? 2. are all distinct 3. in the order rolled, form an increasing consecutive sequence eg. 2, 3, 4
1. 0.4213 2. 0.5556 3. 0.0185 4. 0.1111
6.1 Suppose (P(A) > 0. Find P(B|A) when: 1. B = A 2. B is a superset A 3. B = Omega 4. B = A^c 5. A intersection B = emptyset 6. B = emptyset
1. 1 2. 1 3. 1 4. 0 5. 0 6. 0
6.1 Two balls are painted red or blue uniformly and independently. Find the probability that both balls are red if: 1. At least one is red 2. A ball is picked at random and it is painted red.
1. 1/3 2. 1/2
4.2 Find the number of 7 character (capital letter or digit) license plates possible if no character can repeat and: 1. There are no further restrictions 2. The first 3 characters are letters and the last 4 are numbers 3. letters and numbers alternate
1. 42072307200 2. 78624000 3. 336960000
6.2 An urn contains one red and one black ball. Each time, a ball is drawn independently at random from the urn, and then returned to the urn along with another ball of the same color. For example, if the first ball drawn is red, the urn will subsequently contain two red balls and one black ball. 1. What is the probability of observing the sequence r, b, b, r, r? 2. What is the probability of observing 3 red and 2 black? 3. What is the probability of observing 7 red and 9 black?
1. 1/60 2. 0.16665 3. 0.0588234504
6.5 It rains in Seattle one out of three days, and the weather forecast is correct two thirds of the time (for both sunny and rainy days). You take an umbrella if and only if rain is forecasted. 1. What is the probability that you are caught in the rain without an umbrella 2. What is the probability that you carry an umbrella and it does not rain?
1. 1/9 2. 2/9
4.3 1.In how many ways can you select a group of 2 people out of 5? 2. In how many ways can you select a group of 3 out of 5? 3. In how many ways can you divide 5 people into two group, where the first group has 2 people and the second has 3?
1. 10 2. 10. 3. 10
4.7 1. What is the coefficient of x^3y^2 in expansion of (x + 2y + 1)^10 2. What is the coefficient of x^3 in expansion of (x^2 - x + 2)^10
1. 10080 2. -38400
4.7 How many anagrams, with or without meaning, does "REFEREE" have such that: 1. There is no constraint 2. Two R's are separated 3. It contains subword "EE" 4. It begins with letter "R"
1. 105 2. 75 3. 102 4. 30
4.7 How many anagrams, with or without meaning, do the following words have? 1. CHAIR 2. INDIA 3. SWIMMING
1. 120 2. 60 3. 10080
6.5 A fair coin with P(heads) = 0.5 and a biased coin with P(heads) = 0.75 are placed in an urn. One of the two coins is picked at random and tossed twice. Find the probability: 1. Of observing two heads 2. That the biased coin was picked if two heads are observed
1. 13/32 2. 9/13
3.6 How many ordered pairs (A, B) where A, B are subsets of {1, 2, 3, 4, 5} are there if: 1. A union B {1, 2, 3, 4, 5} 2. |A union B| = 4
1. 132 2. 312
6.3 An urn contains 15 white and 20 black balls. The balls are withdrawn randomly, one at a time, until all remaining balls have the same color. Find the probability that: 1. All remaining balls are white 2. There are 5 remaining balls
1. 15/35 2. 0.03
7.1 An urn contains 20 balls numbered 1 through 20. Three of the balls are selected from the run randomly without replacement, and X denotes the largest number selected. 1. How many values can X take? 2. What is P(X = 18) 3. What is P(X >= 17)
1. 18 2. .119 3. 29/57
4.3 A library has 5 history books, 3 sociology texts, 6 anthropology books, and 4 psychology texts. FInd the number of ways a student can choose: 1. One of the texts 2. Two of the texts 3. One history book and one other type of book 4. one of each type of book 5. two of the books with different types
1. 18 2. 153 3. 65 4. 360 5. 119
2.1 How many elements do the following sets have? 1. {a, b} 2. {{a, b}} 3.{{a, b}, {b, a}, {a, b, a}} 4. {a, b, {a, b}}
1. 2 2. 1 3. 1 4. 3
5.3 A bag contains 5 red and 3 blue balls. 1. Pick one ball at random and observe its random color. What is the size of the color sample space? 2. What is P(blue)? 3. Two balls added to the bag and P(blue) = 0.4. How many of the two balls are blue? 4. Two balls are removed from the original bag and now P(blue) = 0.5. How many of the two balls were blue?
1. 2 2. 3/8 3. 1 4. 0
4.8 How many 6-digit sequences are: 1. Strictly ascending (EG 024579, 135789, but not 011234) 2. Ascending (not necessarily strictly (EG 023689, 033588, or 222222)
1. 210 2. 5005
3.6 How many ordered pairs (A, B), where A, B are subsets of {1, 2, 3, 4, 5} have: 1. A intersect B = \emptyset\ 2. A intersect B = {1} 3. |A intersect B| = 1
1. 243 2. 81 3. 912
Out of 100 foreign journalists who speak Chinese, English, or French at a preference: 60 speak Chinese. 65 speak english 60 speak french 35 speak both C and E 25 speak both Chinese and French 35 speak both english and French. 1. How many journalists speak exactly one language? 2. How many journalists speak two languages? 3. How many speak exactly 3 languages?
1. 25 2. 65 3. 10
4.3 A standard 52-card deck consists of 4 suits and 13 ranks. Find the number of 5 card hands where: 1. Any hand is allowed 2. All five cards are of same suit 3. All four suits are present 4. All cards are of distinct ranks
1. 2598960 2. 5148 3. 685464 4. 1317888
4.8 If a + b + c + d = 10, how many integer solutions (a, b, c, d) are there, when all elements are: 1. non-negative 2. positive
1. 286 2. 84
2.4 Given the expression 'm exists in A', what can be said? 1. m belongs to A 2. A is a member of m 3. m is an element of the set A 4. m is a set of elements 5. A contains m
1. 3. 5
6.3 Eight equal-strength players, including Alice and Bob, are randomly split into 4 pairs, and each pair plays a game, resulting in four winners. Find the probability that: 1. both alice and bob will be amount the 4 winners. 2. Neither alice and bob will be among the 4 winners.
1. 3/14 2. 3/14
6.1 Given events A, B with P(A) = 0.5, P(B) = 0.7, and P(A intersection B) = 0.3. Find: 1. P(A|B) 2. P(B|A) 3. P(A^c | B^c) 4. P(B^c | A^c)
1. 3/7 2. 3/5 3. 1/3 4. 1/5
5.5 In Blackjack, ace is worth 11, face cards worth 10, everything else makes sense. 1. What is the probability that you draw 2 cards and they sum 21 2. What is the probability that you draw 2 cards and they sum 10. 3. Suppose you have drawn two cards: 10 of clubs and 4 of hearts. You now draw a third card from the remaining 50. What is the probability that the sum is > 21
1. 32/663 2.37/897 3. 27/50
5.5 Let X be the number of draws from a deck, without replacement, til an ace is observed. Find: 1. P(X=10) 2. P(X=50) 3. P(X<10)
1. 328/7735 2. 0 3. 4209/7735
4.7 How many ways can we divide 12 people into: 1. three labeled groups evenly 2. three unlabeled groups evenly 3. three labeled groups with 3, 4, and 5 people 4. three unlabeled groups with 3, 4, and 5 people. 5. three unlabeled groups with 3, 3, and 6 people
1. 34650 2. 5775 3. 27720 4. 27720 5. 9240
4.3 Ten points are placed on a plane, with no three on the same line. Find the number of: (Check the question in the quizzes) 1. Lines connecting two of the two points. 2. These lines that do not pass through two specific points (say A or B) 3. Triangles formed by 3 of the points 4. These triangles that contain a given point 5. These triangles contain the side AB
1. 45 2. 28 3. 120 4. 36 5. 8
4.1 In how many ways can three couples be seated in a row so that each couple sits together (namely next to each other): 1. In a row 2. In a circle
1. 48 2. 96
5.5 Find the probability that a 5-card hand contains: 1. The ace of diamonds 2. At least an ace 3. At least a diamond
1. 5/52 2. 0.3412 3. 0.7785
5.3 A standard poker deck has 52 cards, of 13 ranks, and 4 suits. What is the probability that a hand of five cards contains: 1. A queen of hearts. 2. At least one queen 3. At least one heart
1. 5/52 2. 886656/2598960 3. 242784360/311875200
5.6 Let X be distributed over Omega = {1, 2, ..., 100} with P(X = i) = i/k for some integer k. Find: 1. k 2. |E| where E = {x|x exists in Omega, x is multiples of 3} 3. P(E)
1. 5050 2. 33 3. 0.3333
3.5 (Cartesian Powers) Find the number of 7-character (capital letter or digit) license plates possible if: 1. There are no further restrictions 2. The first 3 characters are letter and the last 4 are numbers 3. letters and numbers alternate E.G. A3B5A7Q
1. 78364164096 2. 175760000 3. 632736000
3.5 (Cartesian Powers) Recall that the power set P(S) of a set is the collection of all subsets of S. For A = {1, 2, 3} and B = {x, y} calculate the following cardinalities. 1. |P(A)| 2. |P(B)| 3.|A x B^2| 4. |P(A x B)| 5. |P(A) x B| 6. |P(P(A))|
1. 8 2. 4 3. 12 4. 64 5. 16 6. 256
4.8 In how many ways can we place 10 identical red balls and 10 identical blue balls into 4 distinct urns if: 1. There are no constraints 2. The first urn has at least 1 red ball and at least 2 blue balls. 3. Each urn has at least 1 ball
1. 81796 2. 36300 3. 65094
3.6 Palindrome questions, first digit of an integer palindrome cannot be 0. 1. How many positive 5-digit integer palindromes are there? 2. How many are even, for example 29192? 3. How many contain 7 or 8, for example 27172 or 38783
1. 900 2. 400 3. 452
2.6 Which of the following is not in the cartesian product of {1, 2} x {3, 4} 1. (3, 1) 2. (1, 3) 3. (1, 4) 4. (2, 4)
1. Bc the cartesian product is (1, 3), (1, 4), (2, 3), (3, 4). Note that these are not sets so order matters.
3.2 |A union B union C| = |A| + |B| + |C| whenever: 1. A and B are disjoint and B and C are disjoint: True or False 2. A and B are disjoint, B and C are disjoint, and A and C are disjoint True or False
1. False 2. True
2.5 Which of the following equals G for all Omega and G? 1. G - emptyset 2. Omega - G 3. Omega - G^c 4. G intersect emptyset
1. True 2. False. Omega - G = G^c 3. True 4. False Omega intersect emptyset = emptyset
2.1 List the elements of the following sets 1. {a} 2. {{a}} 3. {a, {b}} 4. {/emptyset/} 5. /emptyset/
1. a 2. {a} 3. a, {b} 4. /emptyset/ 5. none
7.3 If we draw cards from a 52-deck with replacement 100 times, how many times can we expect to draw a black king?
1.923
5.4 You have two fair coins. If you flip a head with the first coin, what is the probability of flipping a head with the second?
1/2
5.2 An outcome in a uniform probability space has probability 1/10. What is the size of the sample space?
10
7.3 Choose a random subset of {2^1, 2^2, ..., 2^10} by selecting each of the 10 elements independently with probability 1/2. Find the expected value of the smallest element in the subset (e.g. the subset can be {2^1, 2^3, 2^4, 2^7}. The smallest element is ).
10
6.3 A box has seven tennis balls. Five are brand new, and the remaining two had been previously used. Two of the balls are randomly chosen, played with, and then returned to the box. Later, two balls are again randomly chosen from the seven and played with. What is the probability that all four balls picked were brand new.
10/147
3.1 A square of an integer, for example, 0, 1, 4, 9 is called a perfect square. How many perfect squares are <= 100?
11
4.1 In how many ways can 11 soccer players form a line before a game?
11!
6.5 An urn labeled "heads" has 5 white and 7 black balls, and an urn labeled "tails" has 3 white and 12 black balls. Flip a fair coin, and randomly select on ball from the "heads" or "tails" urn according to the coin outcome. Suppose a white ball is selected, what is the probability that the coin landed tails?
12/37
4.2 Eight books are placed on a shelf. Three of them form a 3-volume series, two form a 2-volume series, and 3 stand on their own. In how many ways can the eight books be arranged so that the books in the 3-volume series are placed together according to their correct order, and so are the books in the 2-volume series? Noted that there is only one correct order for each series.
120
4.4 Your school offers 6 science classes and 5 art classes. How many schedules can you form with 2 science and 2 art classes if order doesn't matter? 1. 25 2. 55 3. 60 4. 150
150 = (6 choose 2) * (5 choose 2)
2.3 Visualizing Sets A venn diagram for 2 sets has 4 regions, for three sets has 8 regions. How many regions are there in a Venn diagram of 4 sets?
16.
6.5 An ectopic pregnancy is twice as likely to develop when a pregnant woman is a smoker than when she is a nonsmoker. If 32% of women of childbearing age are smokers, what fraction of women having ectopic pregnancies are smokers?
16/33
4.1In how many ways can 8 distinguishable rooks be placed on a 8x8 chessboard so that none can capture any other, namely no row and no column contains more than one rook? For example, in a 2x2 chessboard, you can place 2 rooks labeled 'a' and 'b' in 4 ways. There are 4 locations to place 'a', and that location determines the location of 'b'
1625702400
4.7 In how many ways can you give three baseball tickets, three soccer tickets, and three opera tickets, all general admission, to nine friends so each get one ticket?
1680
5.2 Given a uniform probability space {1, 2, 3, ..., 100}, what is the probability that the outcome contains the digit 1?
19/100
3.5 (Cartesian Powers) If P and Q are sets, then |P| ^ |Q| is the number of functions from: 1. From P to Q 2. From Q to P
2
4.2 How many 2-permutations do we have for set {1, 2, 3, 4}? 1. 8 2. 12 3. 16
2
5.1 Which of the following outcomes are random (not certain) when rolling a six-sided dice? 1. A real number 2. An even number 3. A positive number
2
6.2 Two disjoint events cannot be independent 1. Yes 2. Not exactly
2
6.5 Each of Alice, Bob, and Chuck shoots at a target once, and hits it independently with probabilities 1/6, 1/4, and 1/3, respectively. If only one shot hit the target, what is the probability that Alice's shot hit the target? 1. 31/72 2. 6/31 3. 10/31 4. 15/31
2
7.3 Which of the following statements are true for a random variable X? 1. E(X) must be in the range (0, 1) 2. E(X) can take a value that X does not take 3. P(X <= E(X)) = 1/2 4. E(X) = 1/2(xmax + xmin)
2
2.4 Which of the following are subsets of A = [2, 4) 1. C = {2, 3, 4} 2. D = (2, 4) 3. E = /emptyset/
2 and 3
2.4 Let P(S) be the collection of all subsets of S, and Q(S) be the collection of all proper subsets of S. Which of the following hold for every set of S 1. P(S) is a subset of Q(S) 2. P(S) is a superset of Q(S) 3. P(S) is a proper subset of Q(S) 4. P(S) = Q(S)
2 and 3. The collection of Q(S) is P(S) minus S itself. Hence superset, and proper subset hold, while the rest do not.
5.6 Under which of the following probability assignments does S = {a1, a2, a3} become a probability space? 1. P(a1) = 0.2, P(a2) = 0.3, P(a3) = 0.4 2. P(a1) = 0.2, P(a2) = 0.3, p(a3) = 0.5 3. P(a1) = 0.3, P(a2) = -0.2, P(a3) = 0.9) 4. P(a1) = 0.2, P(a2) = 0, P(a3) = 0.8
2 and 4
5.6 Which of the following always holds? 1. A subset B --> P(A) < P(B) 2. A subset B --> P(A) <= P(B) 3. A isa proper subset B -> P(A) < P(B) 4. A is a proper subset of B -> P(A) <= P(B)
2 and 4
5.6 Which of the following statements are true? 1. If P(E) = 0 for event E, then E = emptyset 2. If E = emptyset, then P(E) = 0. 3. IF E1 union E2 = Omega, then P(E1) + P(E2) = 1. 4. If P(E1) + P(E2) = 1, then E1 union E2 = Omega
2 and 4
6.1 If A and B are disjoint positive-probability events, then P(A|B) = 1. P(A) 2. P(B|A) 3. P(A union B) 4. P(A intersection B)
2 and 4
5.2 Which of the following sample spaces are uniform? 1. {land, sea} for a random point on a globe 2. {odd, even} for a random integer from {1, 2, ..., 100} 3. {leap year, non-leap year} for a random year before 2019 4. {two heads, two tails, one head, and one tail} when flipping two fair coins 5. {distance to origin} for a random point in {-3, -1, 1, 3} x {-4, -2, 2, 4}
2 and 5
7.1 Which of the following are true for random variables? 1. A random variable X defines an event 2. For a random variable X and a fixed real number a, "X <= a" defines an event. 3. Random Variables for the same sample space must be same. 4. For a random variable X, possible values for P(X = x) include 0, 0.5, and 1
2, 3, and 4
2.2 Which of the following are true? 1. e exists in {1, 2, ..., 10} 2. pi exists in (3, 3.5) 3. 2 exists in [-2, 2)
2.
2.2 Which of the following hold? 1. {0} = /emptyset/ 2. {0, 1, 2} = {2, 0, 1, 1} 3. {{0}, 1} = {0, {1}}
2.
2.4 If S is a proper, or strict, subset of T, then: 1. S cannot be empty 2. T cannot be empty 3. S and T must intersect
2.
3.2 We saw that the size of a union of two disjoint sets is the sum of their sizes. If two sets are not necessarily disjoint, then the size of their union is: 1. At least the sum of the set sizes 2. At most the sum of the set sizes 3. Could be smaller, same, or larger than the sum of the set sizes.
2.
3.3 The following equation is incorrect. What needs to be added to make it correct? |A union B union C| = |A| + |B| + |C| - |A intersect B| - |A intersect C| - |B intersect C| 1. -|A intersect B intersect C| 2. + |A intersect B intersect C| 3. + 3|A intersect B intersect C|
2.
3.4 (Cartesian Products) Taking the geometric view of Cartesian products, if A and B are real intervals of positive length in R, then A x B is a: 1. line, 2. rectangle 3. circle 4 triangle
2.
6.2 Of 10 students, 4 take only history, 3 take only math, and 3 take both history and math. If you select a student at random, the event that the student takes history and the event that the student takes math are: 1. Independent 2. Dependent
2.
6.5 Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car and behind the others are goats. You pick a door, say door 1. The host knows what is behind each door. He opens another door, say door 3, which has a goat. He then says to you, "Do you want to change your selection to door 2?" Is it to your advantage to switch your choice? 1. It is better to keep my choice of door 1 2. It is better to switch to door 2 3. There is no differene
2.
6.1 Find the probability that the outcome of a fair-die roll is at least 5, given that it is at least 4.
2/3
5.3 What is the probability of drawing a Red Ace from a standard deck of cards?
2/52. Heart ace and diamond ace?
6.4 Each of Alice and Bob has an identical bag containing 6 balls numbered 1, 2, 3, 4, 5, and 6. Alice randomly selects one ball from her bag and places it in Bob's bag, then Bob randomly select one ball from his bag and places it in Alice's bag. What is the probability that after this process the content in two bags remain unchanged?
2/7
6.3 A box contains six tennis balls. Peter picks two of the balls at random, plays with them, and returns them to the box. Next, Paul picks two balls at random from the box (they can be the same or different from Peter's balls), plays with them, and returns them to the box. Finally, Mary picks two balls at random and plays with them. What is the probability that each of the six balls in the box was played with exactly once?
2/75
3.7 Imagine NBA finals is a series of 5 games. First to 3 wins wins the champion. If the teams are named A and B, then AAA, ABB, and BABAB are three possible win sequences as one team won three games. What is the total number of possible win sequences?
20. TTo see that, we show that there are 10 win sequences ending with A.There is one such sequence of three games: AAA;three sequences of four games: BAAA, ABAA, and AABA;and six sequences with five games: BBAAA, BABAA, BAABA, ABBAA, ABABA, AABBA.Similarly there are ten sequences ending with B, for a total of 20 win sequences.We will later see a systematic way of counting these sequences.
3.3 In a high school grad exam, 70% passed english, 76% pass math and 66% passed both. If 40 examinees failed in both subjects, what is the total number of examinees?
200
1.3 In rolling a fair 6-sided die 1200 times, roughly how many times would you expect to see a 2?
200. Because 1200 x 1/6 = 200. 1/6 is the probability to get a 2.
4.4 How many ordered pairs (A, B), where A, B are subsets of {1, 2, 3, 4, 5} are there if: |A| + |B| = 4
210
4.3 A company employs 4 men and 3 women. How many teams of three employees have at most one woman?
22
7.3 There are 3 classes with 20, 22 and 25 students in each class for a total of 67 students. Choose one out of the 67 students uniformly at random, and let X denote the number of students in his or her class. What is E(X)?
22.5224
3.6 How many 5-digit ternary strings are there without 4 consecutive 0s, 1s, or 2s? For example. 01210 and 11211 are counted. but 20000, 11112, and 22222 are excluded.
228
4.6 What is the coefficient of x^2 in the expansion of (x+2)^4(x+3)^5
23112
4.1 How many permutations does the set {1, 2, 3, 4} have? 1. 9 2. 18 3. 24 4. 36
24 bc 4! = 24
5.3 50% of UCSD students play soccer, 40% play basketball, 30% play both. What is the probability that a random UCSD student does not play any of the two games? 1. 0 2. 0.1 3. 0.4 5 0.6
3
5.3 Six balls are numbered 1, 2, 3, 4, 5, 6. What is the chance that the number on three balls, picked simultaneously and randomly will sum to a multiple of 3? 1. 1/3 2. 1/4 3. 2.5 4. 4/15
3
6.5 Jack has two coins in his pocket, one fair, and one "rigged" with heads on both sides. Jack randomly picks one of the two coins, flips it, and observes heads. What is the probability that he picked the fair coin? 1. 3/4 2. 2/3 3. 1/3 4. 1/4
3
7.2 All cumulative distribution functions are: 1. Continous 2. Left Continous 3. Right Continious 4. None of the above.
3
7.5 Which of the following doesn't hold for all random variables? 1. E[X+2] = E[X] + 2 2. E[2X] = 2E[X] 3. E[X^2] = E[X]^2 4. All of them hold
3
2.6 Which of the following statements hold for all sets A, B, and C? 1. A x B = B x A 2. A intersect (B x C) = (A intersect B) x (A intersect C) 3. A x (B intersect C) = (A x B) intersect (A x C)
3 is the right answer. 1. does not hold when A and B are nonempty and different. 2. does not hold as the left set is a subset of A while the right set is typically not. 3. holds as (x, y) is in the left set iff x exists in A, y exists in B, and y exists in C, iff (x, y) is in the right set.
2.4 Which of the following is not true 1. {red, green, blue} = {blue, red, green} 2. {1, 2, 3} contains 1 3. 2 exists in {all odd integers}
3.
2.5 if Omega = {x, y, z}, then {x, y}^c is: 1. emptyset 2. z 3. {z} 4. {x, y}
3.
3.5 (Cartesian Powers) The set {000, 001, ..., 111} of all 3 bit strings has the following number of subsets: 1. 2^3 2. 2^6 3. 2^8 4. 2^9
3.
3.6 An n-variable Boolean function maps {0, 1}^n to {0, 1}. How many 4-variable Boolean functions are there? 1. 16 2. 256 3. 65,536
3.
4.6 Find the expansion of (x+y)^3 using the stuff in this course: 1. x^3 + y^3 2. x^3 + x^2y + xy^2 + y^3 3. x^3 + 3x^2y + 3xy^2 + y^3
3.
4.7 What is the coefficient of xy in the expansion (x+y+2)^4 1. 12 2. 24 3. 48
3.
4.8 In how many different ways can you write 11 as a sum of 3 positive integers if order matters? 1. 28 2. 36 3. 45 4. None of the above
3.
5.1 Imagine a single experiment where we flip a coin 6 times, and get 'heads, tails, heads, heads, heads, heads'. Which of the following statements hold? 1. The coin is not fair 2. The coins 'tail' probability is 1/6 3. The sequence is an outcome in the sample space. 4. The sample space of the experiment is {head, tail}.
3.
5.7 Let Omega be any sample space, and A, B are subsets of Omega. Which of the following statements are always true? 1. If |A| + |B| >= |Omega|, then P(A union B) = 1. 2. If |A| + |B| >= |Omega|, then P(A) + P(B) >= 1 3. If P(A) + P(B) > 1, then A intersectinon B != 0 4. If P(A) + P(B) > 1, then P(A union B) = 1
3.
3.5 (Cartesian Powers) Let A be a set with size 5. How many strict subsets does A have?
31
3.6 A password consists of 4 or 5 characters. Each an uppercase letter, lowercase letter, or a digit. How many passwords are there if each of the three character types must appear at least once?
312
3.4 (Cartesian Products) How many positive divisors does 2016 have?
36
4.7 How many terms are there in the expansions of (x + y + z)^10 + (x - y + z)^10
36
4.1 In how many ways can 7 men and 7 women can sit around a table so that men and women alternate. Assume that all rotations of a configuration are identical hence counted as just one.
3628800
5.6 Jack solves a Math problem with probability 0.4, and Rose solves it with probability 0.5. What is the probability that at least one of them can solve the problem? 1. 0.7 2. 0.9 3. 0.6 4. Not enough information
4
6.3 A bag contains 4 white and 3 blue balls. Remove a random ball and put it aside. Then remove another random ball from the bag. What is the probability that the second ball is white? 1. 3/6 2. 4/6 3. 3/7 4. 4/7
4
4.6 What is the coefficient of x^2 in the expansion of (x+2)^4 1. 12 2. 24 3. 48
4 choose 2 * 2^2 = 24
2.5 If A - B = A for sets A and B, then: 1. B must be an empty set 2. B must be a subset of A 3. A and B must intersect 4. A and B must be disjoint
4.
2.5 Suppose that the following hold for an unknown set C. Which of the following implies A = B, regardless of what C is? 1. A - C = B - C 2. A intersect C = B intersect C 3. A union C = B union C 4. A symmetric difference C = B symmetric difference C
4.
2.6 Which of the following statements hold? 1. {1, 2} is subset of {1, 2}^2 2. {1, 2} exists in {1, 2}^2 3. (1, 2) is a subset of {1, 2}^2 4. (1, 2) exists in {1, 2}^2
4.
5.4 5 engineers and 3 artists align at random along a line. What is the probability that the first and last are Engineers? 1. 3/14. 2. 8/15 3. 9/14 4. 5/14
4.
5.4 A bag has 3 red and green apples. You start by randomly selecting one red apple from the bag. Which of the following has the highest probability? 1. Select another red apple after replacing the first. 2. Select another red apple without replacing the first. 3. Select a green apple after replacing the first red apple. 4.Select a green apple without replacing the first red apple
4.
5.5 What is the probability that a random 4-card hand consists of a single suit? 1. 4/52 2. 13/52 3. 13c4/52c4 4. (4c1 * 13c4)/52c4
4.
6.4 Three 100-marble bags are placed on a table. One bag has 60 red and 40 blue marbles, one as 75 red and 25 blue marbles, and one has 45 red and 55 blue marbles. You select one bag at random and then choose a marble at random. What is the probability that the marble is red? 1. 0.2025 2. 0.33 3. 0.5 4. 0.6
4.
1.4 What is the probability of drawing a Queen from a deck of 52 cards?
4/52
6.4 Eight equal-strength players, including Alice and Bob, are randomly split 4 into pairs, and each pair plays a game (i.e. 4 games in total), resulting in four winners. What is the probability that exactly one of Alice and Bob will be among the four winners?
4/7
3.3 In a high school graduation exam, 80% examinees passed english, 85% passed math, and 75% passed both. If 40 examinees failed both subjects, what is the total number of examinees?
400
4.1 In how many ways can 8 identical rocks be placed on an 8 x 8 chessboard so that none can capture any other, namely no row and no column contains more than one rook?
40320
6.5 A college graduate is applying for a job and has 3 interviews. She passes the first, second, and third interviews with probabilities 0.9, 0.8, and 0.7, respectively. If she fails any interview, she cannot proceed with subsequent interview(s) and will not get the job. If she didn't get the job, what is the probability that she failed the second interview?
45/124
7.3 A player flips two fair coins. The player wins $3 if 2 heads occur and $1 if head 1 occurs. How much money (in ) should the player lose when no heads occur for the game to be fair (expected gain is 0)?
5
5.4 Roll two fair and distinguishable six-sided dice. What is the probability that the outcome of the second die is strictly greater than the first?
5/12
4.8 How many terms are there in the expansion of (x+y+z)^10
66
4.2 In how many ways can 5 cars, a BMW, Chevy, Fiat, Honda, and a Kia park in 8 spots?
6720
A deck n >= 5 cards has as many 5-card hands as 2-card hands. What is n?
7
How many integers in [1, 100] do not contain the digit 6?
81
4.2 A derangement is a permutation of the elements such that none appear in its original position. For example, the only derangements of {1, 2, 3} are {2, 3, 1} and {3, 1, 2}. How many derangements does {1, 2, 3, 4} have?
9
6.2 4 freshman boys, 6 freshman girls, and 6 sophomore boys go on a trip. How many sophomore girls must join them if a student's gender and class are to be independent when a student is selected at random?
9
1.3 Which of the following describes the differences between probability and statistics? A) Probability predicts what will happen. Statistics, in part, uses what has already happened. B) Probability requires existing data. Statistics requires underlying models. C) THey're the same thing.
A
2.2 Which of the following define a set unambiguously? A) {3, 4, 5, 7} B) {negative primes} C) {good drivers in San Diego}
A & B
1.1 What are probability and statistics useful for? A) Quantifying Uncertainty B) Finding exact solutions to mathematical equations. C) Making Predictions about the future.
A & C There is no uncertainty in B.
2.1 Which of the following hold? A) 0 exists {0, 1} B) a exists {A, B} C) {a, b} exists {{a, b}, c}
A and C
3.1 The Python definition A = set(range(1, 10)) implies that A has size: 1. 2 2. 9 3. 10 4. 11
A has 9 elements as the elements are 1 to 9
1.1 Which of the following are best solved using probability and statistics? A) Predicting the number of rainy days in April B) Approximating the closing price of IBM stock tomorrow. C) Estimating your potential winnings in a game of Blackjack. D) Guessing the winner of the next World Cup.
A, B, C, D
1.2 When the number of coin flips increases, the distribution of the sum of Heads (1) and Tails (-1) becomes: A) More concentrated around zero B) Skewed to positive values C) Skewed to negative values D) More spread out across all possibilities
A. The randomness will even out, and what we get is the average of -1 and 1 which is 0.
2.1 Let A be the set of anagrams of singular English animal names. For example, "nails" and "slain" are anagrams of "snail". So all three exist in A, yet "bar" does not exist in A. Which of the following exist in A? 1. tan 2. pea 3. low 4. bare 5. loin 6. bolster
All of the above
2.5 Which of the following imply A = B 1. A^c = B^C 2. A symmetric difference B = emptyset 3. A - B = B - A 4. A union B = A intersect B
All of the above
3.3 |A Union B| = |A| + |B| when: 1. A and B are disjoint 2. A is the complement of B 3. A and B do not intersect 4. At least one of A and B is empty
All of the above
3.3 When does |A union B| = |A| + |B|? 1. When at least one of A and B is empty? 2. When A and B are disjoint 3. Both of the above.
Both of above.
2.1 (T or F) The universal set (Omega) is unique
False
7.3 The expectation of a random variable X must be a number X can take: True or False
False
2.2 How many elements are there in the real interval [2, 4)
Infinitely many points: 2, 2.1, 2.11, 2.111, etc
1.3 If we flip a coin a thousand times and get 507 heads, can we conclude with certainity that the coin is unbiased?
No
5.6 Does P(A) = 0 imply that A is the empty set?
Not necessarily
7.3 A quiz-show contestant is presented with two questions, question 1 and question 2, and she can choose which question to answer first. If her initial answer is incorrect, she is not allowed to answer the other question. If the rewards for correctly answering question 1 and 2 are $200 and $100 respectively, and the contestant is 60% and 80% certain of answering question 1 and 2, which question should she answer first as to maximize the expected reward?
Question 2
1.1 (T or F) Probability and Statistics provide mathematical tools for estimating the likelihood of random events
True
1.4 (True or False) If we repeat an experiment many times, the long-term frequencies of the outcomes converge to the probabiilities.
True
2.1 (T or F). The empty set is unique
True
6.1 Let A and B be two positive-probability events. Does P(A|B) > P(A) imply P(B|A) > P(B)?
Yes
7.4 Let X be a random variable. For a fixed real function g, g(X) is also a variable. Yes or no?
Yes
1.3 A coin is tossed 1000 times and turns up heads 700 times. Is the coin biased?
Yes. The probability that an ubiased coin would generate 700 heads is small. Hence we can be confident that it is biased.
4.5 For a positive integer n, n choose (n-1) equals to: 1. 1 2. n-1 3. n 4. n+1
n
4.5 if (n+2)C5 = 12(nC3). find n
n = 14
4.1 Which of the following are true for all n, m exists in N, and n >= 1 1. n! = n * (n-1)! 2. (n*m)! = n! * m! 3. (n + m)! = n! + m! 4. (n^m)! = (n!)^m
n! = n * (n-1)!
