Probability Exam #1 Homework Questions

The distribution of blood types in the United States according to the ?ABO classification? is O: 45%, A:40%, B: 11%, and AB: 4%. Blood is also classified according to Rh type, which can be negative or positive and is independent of the ABO type (the corresponding genes are located on different chromosomes). In the U.S. population, about 84% are Rh positive. Sample two individuals at random and find the probability that at least one of them is O positive.

((.45)(.84))^2+(1-(.45)(.84))

Three numbers 1-n are chosen at random. What is the probability that the smallest number is j and the largest is k?

((k-j + 1) C 3)/(n C 3)

The distribution of blood types in the United States according to the ?ABO classification? is O: 45%, A:40%, B: 11%, and AB: 4%. Blood is also classified according to Rh type, which can be negative or positive and is independent of the ABO type (the corresponding genes are located on different chromosomes). In the U.S. population, about 84% are Rh positive. Sample two individuals at random and find the probability that both are A negative.

(.4)(.16)(.4)(.16)

The distribution of blood types in the United States according to the ?ABO classification? is O: 45%, A:40%, B: 11%, and AB: 4%. Blood is also classified according to Rh type, which can be negative or positive and is independent of the ABO type (the corresponding genes are located on different chromosomes). In the U.S. population, about 84% are Rh positive. Sample two individuals at random and find the probability that they have the same ABO type.

(.4)^2+(.11)^2+(.45)^2+(.04)^2

You roll a die and flip a fair coin a number of times determined by the number on the die. What is the probability that you get no heads?

(1/6)(sum from 1 to 6 of 1/2^k)

What is the probability of getting four of a kind?

(13 C 1) * 12 * 4/(52 C 5)

What is the probability of getting a full house?

(13 C 1)(4 C 3)(4 C 2)(12 C 1)/(52 C 5)

An Indiana license plate consists of three letters and three digits. What is the probability that the plate has no duplicate letters?

(26 * 25 * 24)(10^3)/(26^3)(10^3)

An Indiana license plate consists of three letters and three digits. What is the probability that the plate has no duplicate letters and all digits equal?

(26*25*24)(10)/(26^3)(10^3)

An Indiana license plate consists of three letters and three digits. What is the probability that the plate has no duplicate digits?

(26^3)(10 * 9 * 8)/(26^3)(10^3)

An Indiana license plate consists of three letters and three digits. What is the probability that the plate has only odd digits?

(26^3)(5^3)/(26^3)(10^3)

On a chessboard, you place three pieces at random. What is the probability that they are all on black squares?

(32 C 3)/(64 C 3)

Suppose there are four envelopes containing different amounts of money. What is the probability of getting the largest amount when taking the first envelope?

(4 C 1) = 1/4

What is the probability of getting a flush?

(4 C 1)(13 C 5)/(52 C 5)

What is the probability of getting a straight flush?

(4 C 1)*10/(52 C 5)

What is the probability of getting a royal flush?

(4 C 1)/(52 C 5)

Three fair dice are rolled. Given that there are no 6s, what is the probability that there are no fives?

(4/5)^3

Three numbers 1-10 are chosen at random. What is the probability that the smallest number is 4 and the largest is 8?

(5 C 3)/(10 C 3)

Three numbers 1-10 are chosen at random. What is the probability that the smallest number is 4?

(7 C 3)/(10 C 3)

On a chessboard, you place three pieces at random. What is the probability that they are all in the first row?

(8 C 3)/(64 C 3)

Regular coordinate system--j steps sideways, k steps upwards (j + k = n total steps). What is the probability that all the j steps came in a row?

(k+1)/(n C j)

What is the probability that Bob's luggage was mishandled at LAX given that his luggage was missing in Sydney?

1 = P(L|M) + P(O|M) P(L|M) = 1 − 1/(2-p) ≈ 0.4975

A city with 10 districts has 8 robberies in one week. What is the probability that some district had more than one robbery?

1-(10!/10^8)

A city with 6 districts has 6 robberies in one week. What is the probability that some district had more than one robbery?

1-(6!/(6^6))

Roll a die 10 times. What is the probability of getting at least two 6s?

1-(P(X=0)+P(X=1))

The random variable X has a binomial distribution with E[X] = 1 and var[X] = 0.9. Compute P(X>0).

1-P(X=0) = 1-(.1)^0(.9)^10

Regular coordinate system--j steps sideways, k steps upwards (j + k = n total steps). What is the probability that all the j steps came first?

1/(n C j)

The random variable X has pmf p(k) = c/2^k. Find the constant C, P(X>0), and the probability that X is even.

1/2^k converges to 2, so c(2)=1, c = 1/2. P(X>0) 1-P(X = 0) = 1/2 P(even)= convergent series: 1/2 (1 + 1/2^2 + 1/2^4...)= 1/2 (1/1-1/4) = 2/3

On a chessboard, you place three pieces at random. What is the probability that they are all in the same row and on the same color?

2 * 8 * (4 C 3)/(64 C 3)

The distribution of blood types in the United States according to the ?ABO classification? is O: 45%, A:40%, B: 11%, and AB: 4%. Blood is also classified according to Rh type, which can be negative or positive and is independent of the ABO type (the corresponding genes are located on different chromosomes). In the U.S. population, about 84% are Rh positive. Sample two individuals at random and find the probability that one of them is O and Rh positive, while the other is not.

2(.45)(.84)(1-(.45)(.84))

They have the same ABO type and different Rh types.

2(.84)(.16)((.4)^2+(.11)^2+(.45)^2+(.04)^2)

The distribution of blood types in the United States according to the ?ABO classification? is O: 45%, A:40%, B: 11%, and AB: 4%. Blood is also classified according to Rh type, which can be negative or positive and is independent of the ABO type (the corresponding genes are located on different chromosomes). In the U.S. population, about 84% are Rh positive. Sample two individuals at random and find the probability that one is Rh positive and the other is not AB.

2(0.96)(.84)-((.96)(.84))^2

Regular coordinate system--j steps sideways, k steps upwards (j + k = n total steps). What is the probability that all the j steps came before or after the k steps up?

2/(n C j)

A password must consist of 5 letters and 2 digits. How many possible passwords are there if repetitions are allowed?

21 * (26^5) * (10^2)

A password must consist of 5 letters and 2 digits. How many possible passwords are there if repetitions are not allowed?

21 * 26_5 * 10_2

An Indiana license plate consists of three letters and three digits. What is the probability that the plate has all letters the same?

26*(10^3)/(26^3)(10^3)

What is the probability of typing the word HAMLET in 10 letters if you are not allowed to repeat letters?

5*(20_4)/(26^10)

What is the probability of typing the word HAMLET in 10 letters if you are allowed to repeat letters?

5*(26^4)/(26^10)

On a chessboard, you place three pieces at random. What is the probability that they are all in the same row?

8 * (8 C 3)/(64 C 3)

Roll a die 10 times. What is the probability of getting no 6s?

Bin(10,1/6)

Roll a die 10 times. What is the probability of getting at most three 6s?

Bin(10,1/6) P(X = 3) + P(X = 2) + P(X =1) + P(X = 0)

Five cards are drawn at random from a deck of cards. Let X be the number of aces. Find the pmf of X if the cards are drawn with replacement.

Bin(5,4/52)

Suppose there are four envelopes containing different amounts of money. What is the probability of getting the largest amount when discarding the first envelope and taking the next amount higher?

Calculate probabilities of picking all the combinations--1st highest, then 4th highest, (etc), 1st highest then second highest then 4th highest, (etc), 1st highest then second highest then third highest then fourth highest, add them together

A: 1,1,5,5,5,5 B: 3,3,3,4,4,4 C: 2,2,2,2,6,6 Which die is most likely to win?

Calculate relevant probabilities: A: P(1>B, 1>C)P(1) + P(5>B,5>C)P(5)

Find the expected number of different birthdays amongst four people selected at random.

E[B] = 1 * P(B = 1) + 2 * P(B = 2) +...

Let X be the number of 6s when a die is rolled six times, and let Y be the number of 6s when a die is rolled 12 times. The expectation of X and Y, the probability that X > E[X], and the probability that Y > E[Y].

E[x] = 1 E[y] = 2 P(X>E[x]) = P(X > 1) P(Y>E[y])= P(Y>2)

A drunken man has 5 keys, one of which opens the door to his office. He tries the keys at random, one by one and independently. Compute the expectation and variance of the number of tries required to open the door if the wrong keys are not eliminated.

Geom(1/5)

Five cards are drawn at random from a deck of cards. Let X be the number of aces. Find the pmf of X if the cards are drawn without replacement.

HGeom(w=4, b= 48)--calculate relevant probabilities: ex. P(X=1)= (4C1)(48C4)/(52C5)

Roll two dice and find the pmf of X if X is the smallest number.

Just write out the combinations, and find all the groupings for each number being the smallest

A roulette wheel is spun eight times. If any of the numbers is repeated, you lose $10, otherwise you win $10. Should you play this game?

Number of possible outcomes is 38^8. Number of possible outcomes without repeats is 38_8, so the probability it 38_8/38^8.

An urn contains n red balls, n red balls, and n black balls. You draw k balls at random without replacement. Find an expression for probability that you don't get all the colors.

P (no red or no black or no white) = 3*P(no red)-3*(no red or white) 3*(2n C k) - 3*(n C k)/(3n C k)

1 gross = 12 dozen. Suppose a gross has 8 cracked eggs. What is the probability that it is accepted?

P(0) = (8 C 0)(136 C 12)/(144 C 12) P(1) = (8 C 1)(136 C 11)/(144 C 12) P(2) = (7 C 0)(125 C 12)/(132 C 12)

Law of Independence

P(A and B) = P(A)*P(B)

A coin has probability p of showing heads. Flip it three times and consider the events A, at most one tails, and B, all flips are the same. For which values of p are A and B independent?

P(A and B) = P(A)*P(B) P(A) = p^3 +3(p^2)(1-p) P(B) = p^3 + (1-p)^3 P(A and B) = P(A)*P(B) p^3 = P(A)P(B) p = 1/2

A politician considers running for election. He has about a 60% chance of winning the first election, a 50% chance of winning the second election. But if he wins the first, there is a 75% chance of winning the second. Find the probability he wins the first but not the second.

P(A but not B) = P(A) - P(A and B) = .15

If he wins the second, what is the probability that he won the first?

P(A given B) = P(A and B)/P(B) = .90

You roll a die and consider the events "the number you get is even" and "you get at least 2". Find P(B given A) and P(A given B)

P(A) = 1/2 P(B) = 5/6 P(B given A) = 1 P(A given B) = 3/5

We sample with replacement a regular deck of cards until we get an ace, or we get a spade but not the ace of spades. What is the probability that the ace comes first?

P(A)/P(A)+P(S) (4/52)/(4/52)+(12/52)= 1/4

Roll two fair dice. Let Ak be the event that the first die gives k and let Bn be the event that the sum is n. For which values of n and k are Ak and Bn independent?

P(Ak and Bn) = P(Ak)P(Bn) Write out the values and count!!

A politician considers running for election. He has about a 60% chance of winning the first election, a 50% chance of winning the second election. But if he wins the first, there is a 75% chance of winning the second. Find the probability he wins both elections.

P(B and A) = P(B given A)P(A) = .45

If he loses the first, what is the probability he wins the second?

P(B given !A) = P(!A and B)/P(!A) = P(!A given B)P(B)/P(!A) = P(B) - P(A and B)/(1-P(A)) = .125

Suppose the probability that an individual is born in the winter half is p. What is the probability that two people are born in the same half of the year? For which value of p is this minimized?

P(both in winter) = p^2 P(both in summer)= (1-p)^2 P(both in same half)=p^2 + (1-p)^2 P(minimized)=take derivative and solve equal to zero.

An object is hidden randomly in one of ten covered boxes numbered from 1 to 10. You search for it by randomly lifting the lids. Find the expected number of lids you need to lift until you locate the object.

P(success for each box) = 1/10, calculate expected value

A drunken man has 5 keys, one of which opens the door to his office. He tries the keys at random, one by one and independently. Compute the expectation and variance of the number of tries required to open the door if the wrong keys are eliminated.

P(success for each key = 1/5), calculate expected value

In the United States, the overall chance that a baby survives delivery is 99.3%. For the 15% that are delivered by cesarean section, the chance of survival is 98.7%. If a baby is not delivered by C section, what is its survival probability?

P(survival)=P(survival given C)P(C)+(survival given !C)(P!C) Solve for (survival given !C)

What does disjoint mean?

Probability of A and B = 0.

The demand for a certain weekly magazine at a newsstand is a random variable with pmf p(i) = (10 − i)/18, i = 4, 5, 6, 7. If the magazine sells for $a and costs $2a/3 to the owner, and the unsold magazines cannot be returned, how many magazines should be ordered every week to maximize profit in the long run?

Profit per magazine = a/3 P(X=4): 4*a/3 P(X=5): Calculate expected value if you sell four and don't sell the fifth one (take a loss), and then if you sell all five (take the probability function * expected profit for each)

A fair coin is flipped twice. Let x be the number of heads minus the number of tails. Find the pmf and sketch the cdf of X.

Range: {-2,0,2} P(X = 2) = 1/4 P(X = -2) = 1/4 P(x = 0) = 1/2

We are given 20 urns each containing 19 balls (19 green, 1 red 18 green, etc.) We select an urn at random and we sample without replacement 2 balls. What is the probability that the second ball is green?

The probability of picking each urn is 1/20. The probability of picking a green ball in each urn is (k-1)(20-k)+(20-k)(19-k)/(19*18)=(20-k)/19. So, when you multiply these together, you get 1/2.

You roll a die twice and record the largest number. Given that the first roll gives a 1, what is the conditional probability that the largest number is 3?

The rolls are independent, so this probability is only that you would roll a 3 on the second roll--1/6

You roll a die twice and record the largest number. Given that the first roll gives a 3, what is the conditional probability that the largest number is 3?

This is the probability that what you roll is less than or equal to three, so 1/2.

Suppose there are four envelopes containing different amounts of money. What is the probability of getting the largest amount when discarding the first two envelopes and taking the next amount that's higher?

Two picks: probability = 1/4*3 Three picks: probability = 1/4*3*2 Four picks: probability = 1/4*3*2*1

What is the probability that Bob's luggage was mishandled at O'Hare given that it was missing in Sydney?

We want to compute P(O|M). We have P(O|M) = P(M|O)P(O), P(M) = P(M|O)P(O) + P(M|Oc)P(Oc). P(M ) Note that P(M|O) = 1, P(M|Oc) = p. Hence P(M)=p+p(1−p), P(O|M)= p/(p + p(1-p)) = 1/(2-p) ≈0.5025.

Roll two dice and find the pmf of X if X is the difference between the largest and the smallest numbers.

Write out all the differences between numbers in the combinations, and assign probabilities to them.

The game of chuck-a-luck is played with three dice, rolled independently. You bet one dollar on one of the numbers 1 through 6 and, if exactly k of the dice show your number, you win k dollars k = 1, 2, 3 (and keep your wagered dollar). If no die shows your number, you lose your wagered dollar. What is your expected loss?

X is a binomial with Bin(3, 1/6). Calculate the expected value of each thing happening--for example, P(X = k) = (3 C K)(1/6)^k(5/6)^3-k and then multiply by each payoff

A urn contains n white and m black balls. What is the probability that the first black ball comes at the kth draw (without replacement)?

m * (n_k-1)/(n+m)_k