Counting Methods and Probability Review-GRE
The probability of rolling a 3 four consecutive times on a six-sided die would be
(1/6)(1/6)(1/6)(1/6) = 1/1296
A certain coin with heads on one side and tails on the other has a 1/2 probability of landing on heads. If the coin is flipped 5 times, how many distinct outcomes are possible if the last flip must be heads? Outcomes are distinct if they do not contain exactly the same results in exactly the same order
(2)(2)(2)(2)(1)=16 Every time there are two options, but the last time there is only one option-getting a head.
A history exam features five questions. Three of the questions are multiple-choice with four options each. The other two questions are true or false. If Caroline selects one answer for every question, how many different ways can she answer the exam?
(4)(4)(4)(2)(2)
An italian restaurant boasts 320 distinct pasta dishes. Each dish contains exactly 1 pasta, 1 meat, and 1 sauce. If there are 8 pastas and 4 meats available, how many sauces are there to choose from?
(8)(4)(x)=320 x=10
An Italian restaurant boasts 320 distinct pasta dishes. Each dish contains exactly 1 pasta, 1 meat, and 1 sauce. If there are 8 pastas and 4 meats available, how many sauces are there to choose from?
(8)(4)(x)=320 x=10
An integer is randomly chose from 2 to 20 inclusive. What is the probability that the number is prime?
*** easy trick. 2 to 20 does not include #1, so it is out of 19 numbers, not 20 numbers. SO, 8/19 NOT 8/20
In a spelling contest, the winner will receive a gold metal, the second-place finisher will receive a silver medal, the third-place finisher will receive a bronze metal, and the fourth place finisher will receive a blue ribbon. If there are 7 entrants in the contest, how many different arrangements of award winners are there?
--Notice the word "arrangements" HINT this is a permutation problem. The gold metal can be won by any 7 people The silver metal can be won by any of the remaining 6 The bronze by remaining 5 Blue ribbon by remaining 4 (7)(6)(5)(4)=840
The probability that Maria will eat breakfast on any given day is 0.5. The probability that Maria will wear a sweater on any given day is 0.3. The two probabilities are independent of each other Quantity A: The probability that Maria eats breakfast or wears a sweater Quantity B: 0.8 Quantity C Quantity D
1. Find the probability of both events occurring (breakfast and sweater) =0.5(0.3) =0.15 The probability of them occurring = 0.5+0.3 =0.8 Subtract 0.8-0.15 = 0.65 |A ∪ B| = |A| +|B| - |A ∪ B|
A bag contains 6 black chips numbered 1-6 respectively and 6 white chips numbered 1-6 respectively. If Pavel reaches into the bag of 12 chips, one after the other, without replacing them, what is the probability that he will pick black chip #3 and then white chip #3?
1/132 The probability of picking a black chip #3 is 1/12. Once Pavel has removed the first chip, only 11 chips remain, so the probability of picking white chip #3 is 1/11. Multiply 1/12x1/11=1/132
The probability is 1/2 that a certain coin will turn up heads on any given toss and the probability is 1/6 that a number cube with faces numbered 1 to 6 will turn up any particular number. What is the probability of turning up a heads and a 6?
1/2 X 1/6 =1/12
A certain coin with heads on one side and tails on the other has a 1/2 probability of landing on heads. If the coin is flipped three times, what is the probability of flipping 2 tails and 1 head, in any order?
1/2(1/2)(1/2) =1/8 But since these can be in any order of the 3 you multiply 1/8 by 3 = 3/8
A bag contains 6 red chips numbered 1 through 6 respectively, and 6 blue chips numbered 1 through 6 respectively. If 2 chips are to be picked sequentially from the bag of 12 chips, without replacement, what is the probability of picking a red chip and the a blue chip with the same number?
1/22 In the question asked, there are six possible ways to fulfill the requirements of the problem, not one, because the problem does not specify whether the number should be 1,2,3,4,5 or 6. Thus, any of the 6 red chips is acceptable for the first pick. However, on the second pick, only the blue chip with the same number as the red one that was just picked is acceptable 6/12x1/11= 1/2x1/11= 1/22
Pedro has a number cube with 24 faces and an integer between 1 and 24 on each face. Every number is featured exactly once. When he rolls, what is the probability that the number showing is a factor of 24?
1/3 You can count 24 in this 24/1 24/24 24/2 24/3 24/4 24/6 24/8 24/12
What is the probability of rolling a six on two consecutive rolls of a fair six-sided die?
1/6 (1/6) = 1/36 This is because there are independent of each other, so the probability of rolling a 6 on the first try is 1/6 and the probability of rolling a 6 on the second try is 1/6 then multiply them together
The probability of rain is 1/6 for any given day next week. What is the probability that it will rain on both Monday and Tuesday?
1/6 X 1/6 = 1/36 This is because it tells you that the probability of it raining is 1/6. since it is for both monday and tuesday, you take both days separately.
A 10 student class is to choose a president, vice president, and secretary from the group. If no person can occupy more than one post, in how many ways can this be accomplished?
10x9x8= 720
A history exam features 5 questions. Three of the questions are multiple choice with four options each. The other two questions are true or false. If Caroline selects one answer for every question, how many different ways can she answer the exam?
256 This question tests the fundamental counting principle, which states that the total number of choices is equal to the product of the independent choices. The five separate test questions means there are fie independent choices. For the three multiple choice questions there are four options each, whereas for the two true/false questions there are two options each. Multiplying the independent choices yields: (4)(4)(4)(2)(2) = 256
If a website calls for a 3-letter password but no two letters can be the same, the total possibilities would be?
26X25X24 =15600 This is because the stipulation that no two letter be the same reduces the number of choices for the second and third letters.
A menu has 3 choices for soup and 4 salad options; diners are permitted to select a soup or salad with their dinners. In this situation, the total number of choices available is?
3+4=7 OR in a sentence means to ADD AND in a sentence means to MULTIPLY
Tarik has a pile of 6 green chips numbered 1 through 6 respectively and another pile of 6 blue chips numbered 1 though 6 respectively. Tarik will randomly pick 1 chip from the green pile and 1 chip from the blue pile. Quantity A: The probability that both chips selected by Tarik will display a number less than 4 Quantity B: 1/2 Quantity C Quantity D
3/6X3/6 =9/36 =1/4 Quantity B
If the word WOW can be rearranged into 3 ways (WOW, OWW, WWO), how many different arrangements of the letters in "MISSISSIPPI" are possible?
34,650 Total number of items! -------------------- First group! Second group! Etc. Because Mississippi has 11 letters, including 1 M, 4 S's, 4 I's, and 2 P's: 11! ---- 1!4!4!2!
If a ballot offers 3 candidate choices for Office A, 4 for Office B, and 2 for Office C, the total number of different ways that a voter could fill out the ballot is
3X4X2 = 24
A man has 3 different suits, 4 different shirts, 2 different pairs of socks, and 5 different pairs of shoes. If an outfit consists of exactly 1 suit, 1 shirt, 1 pair of socks, and 1 pair of shoes, how many different outfits can be made with the man's clothing?
3x4x2x5 =120
A restaurant menu has several options for tacos. There are 3 types of shells, 4 types of meat, 3 types of cheese, and 5 types of salsa. How many distinct tacos can be ordered assuming that any order contains exactly one of each of the above choices?
3x4x3x5
100 tiles are labeled with the integers from 1 to 100 inclusive; no numbers are repeated. If Alma chooses one tile at random, replaces it in the group, and chooses another tile at random, what is the probability that the product of the two integer values on the tiles is odd?
50 are odd 50/100 =1/2 1/2(1/2) = 1/4
How many five digit numbers can be formed using the digits: 5,6,7,8,9,0
6 digits 6! 6(5)(4)(3)(2)(1) =720 --this is the answer because it doesn't specify that digits cannot be repeated
How many 10-digit numbers can be formed using only the digits 2 and 5? A) 2¹⁰ B) (22)(5!) C) (5!)(5!) D)10!/2 E) 10!
A
A state issues automobile license plates that begin with two letters selected from a 26-letter alphabet, followed by four numerals selected from the digits 0 through 9, inclusive. Repeats are permitted. For example, one possible license plate combination is GF3352 Quantity A: The number of possible unique license plate combinations Quantity B: 6,000,000
A The best way to go about this problem is to make 6 slots on paper and fill in the probability for each slot and then multiply them all Ie: - - - - - - We know that things can be repeated, so the probability of getting letters is 26. We also are told that 2 letters are used in license plates 26 (26) Now we have 4 slots to fill in the numbers, and these can be repeated too, so there is the probability of 10 each time. 26(26)(10)(10)(10)(10) = A A is larger than B
A bag contains only red and blue plastic chips. There were 10 chips in the bag and 1 blue chip was removed. The probability of drawing a blue chip was then 1/3 . How many red chips were in the bag?
After 1 blue chip was removed there were 9 chips left. If the probability of drawing another blue chip was then 1/3, there must have been 1/3 (9)=3 blue chips, and 9-3=6 red chips remaining. Since no red chips were drawn, the original number of red chips must also have been 6.
In a class of 25 students, each student studies either Spanish, Latin, or French, or two of the three, but no students study all three languages. 9 study Spanish, 7 study Latin, and 5 study exactly two languages. Quantity A: The number of students who study French Quantity B: 14 Quantity C Quantity D
C 25= 9 + 7 ---- - 5 25=11 14=14
The probability of rain is 1/2 on any given day next week Quantity A: The probability that it rains on at least one of the 7 days next week Quantity B: 127/128 C D
C The probability is given: 1/2 so 1/2x1/2x1/2x1/2x1/2x1/2x1/2 = 1/128
In a class of 25 students, each student studies either Spanish, Latin, or French, or two of the three, but no students study all three languages. 9 study spanish, 7 study latin, and 5 study exactly two languages Quantity A: The number of students who study French B: 14 C D
C The problem specifies that no one studies all three languages. In addition, a total of 5 people study two languages. Thus, 5 people have been double-counted. Since the total number of people who have been double-counted use this formula Total=spanish+french+latin-(two of the three)-2(All three) 25= 9 +F +7-5-2(0) F=14
What is the probability of tossing a fair coin four consecutive times and having the coin land heads up 0,2,3, or 4 times?
Coin heads up: Probability= number of desired outcomes/number of possible outcomes number of possible outcomes: Since a coin is tossed four times and each toss has two possible outcomes, the total number of outcomes is 2⁴ =16 The number of desired outcomes is 4 So 4/16 =1/4 To find the probability of the coin landing heads up not exactly once, subtract that probability from 1: 1-0.25 = 0.75
Jan and 5 other children are in a classroom. The principal of the school will choose two of the children at random. What is the probability that Jan will be chosen?
Denominator= (6)(5) =30 This is because the first try is 6 possible and the second is 5 possible. 30 is then divided by 2 because there are two ways in which it can happen. Say Jan is picked the first time OR say she is picked the second time. The numerator is 5 because there are five children other than Jan 5/15 =1/3
Two number cubes with six faces numbered with the integers from 1 through 6 are tossed. What is the probability that the sum of the exposed faces on the cubes is a prime number?
Don't forget that these can go up to as high as 11 because you can add the 6 and 5 So prime 2,3,5,7,11 2: (1+1) 3: (1+2)(2+1) 5: (1+4)(4+1)(2+3)(3+2) 7: (1+6)(6+1)(4+3)(3+4)(5+2)(2+5) 11: (6+5)(5+6) Thats a total of 15/36 5/12
A bag contains 3 red, 2 blue, and 7 white marbles. If a marble is randomly chosen from the bag, what is the probability that it is not blue?
First, count the amount of marbles: 12 marbles total Find the probability that it will be blue: 2/12=1/6 1-1/6 = 5/6 marbles that are not blue
How many five-digit numbers can be formed using the digits 5,6,7,8,9,0 if no digits can be repeated?
Here, it does specify that the digits cannot be repeated. For the first digit, there are only five options (5,6,7,8 and 9) because 0 cannot be the start of the number sequence since it has to be "five digits" For the second digit, there are 5 choices again, because now zero can be used but one of the other numbers has already been used, and numbers cannot be repeated. For the third number, there are four choices, for the fourth, 3 and so on 5(5)(4)(3)(2)=600 600
How many ways are there to fill a candelabra with 4 candle holders from a box of 6 distinctly colored candles?
Here, order does matter since the candles are distinctly colored and being placed into slots on the candelabras. There are 6 possible candles for the first slot, 5 for the second, 4 for the third, 3 for the fourth. 6x5x4x3 =360
What does it mean by "A is a subset of B"
If all the elements of set A are among the elements of set B, then A is a subset of B.
A number is randomly chosen from a list of 10 consecutive positive integers. What is the probability that the number selected is greater than the average (arithmetic mean) of all 10 integers
In a list of 10 consecutive integers, the mean is the average of the 5th and 6th numbers. Therefore, the 6th through 10th integers is greater than the mean. Since probability is determined by the number of desired items divided by the total number of choices, the probability that the number chosen in greater than the average of all 10 integers is 5/10 or 1/2
How many groups can be formed consisting of 2 people from room A and 3 people from room B if there are 5 people in room A and 6 people in room B?
Insert the appropriate numbers into the combination formula for each room and then multiply the results: For room A: 5! --- 2! (5-2)! (5) (4) (3) (2) (1) -------------- (2)(1) (3) (2) (1) Can reduce to get (5)(4)/(2) =10 For room B: (6)(5)(4)(3)(2)(1) -------------- (3)(2)(1)(3)(2)(1) =20 Finally 10(20)=200 possible groups
In the GRE, you can expect "the probability of A or B" to be the same as "the probability of A or B or both" What is the equation used?
P (A or B)= P (A) + P (B) - P (A and B)
"How many arrangements are possible"
Permutation
"How many orders are possible"
Permutation
"How many schedules are possible"
Permutation
"How many ways are possible"
Permutation
The 4 finalists in a spelling contest win commemorative plaques. If there are 7 entrants in the spelling contest, how many possible groups of finalists are there?
Plug the numbers into the combination formula, such that n is 7 (the number in the large group) and k is 4 (the number of people in each subgroup formed) n!/k! (n-k)! 7! ---- 4! (7-4)! At this stage, it is helpful to reduce these terms. Since 7 factorial contains all the factors of 4 factorial, we can write 7! as (7)(6)(5) and then cancel the 4! in the numerator and denominator (7)(6)(5) ------- (3)(2)(1) Answer=35
If all possible outcomes of the experiment are equally likely to occur, what is the probability equation?
Probability= number of desired outcomes/number of possible outcomes
How do you find the number of elements in a union of two sets?
Remember, a union is the area where the two things intersect (like in a Venn Diagram). |A ∪ B| = |A| +|B| - |A ∪ B|
Paula has 10 books that she'd like to read on a vacation, but she only has space for 3 books in her suitcase. How many different groups of 3 books can Paula pack?
Since the books are just being put in a suitcase, the order doesn't matter, and the combinations formula can be used 10! ---- 3! (10-3) When canceling, you are left with 10X9X8 ------- 3X2 = 720/6 = 120
A number is randomly chose from the first 100 positive integers. What is the probability that it is a multiple of 3?
The first 100 positive integers comprise the set of numbers containing the integer 1 to 100. Of these numbers, the only ones that are divisible by 3 are (3,6,9,12,15,18,21,24,27,30,33...) which adds up to exactly 33 numbers. this can be determined in several ways. One option is to count the multiples of 3, but that's a bit slow. Alternatively, compute 99/3=33 and realize that there are 33 multiples of 3 up to and including 99. The number 100 is not divisible by 3, so the correct answer is 33/100
A and B are overlapping sets. If |A| has 7 elements, |B| has 5 elements and |A∩B| has 3 elements, how many elements are in |A∪B|?
The formula based on the inclusion-exclusion principle for sets states that |A∪B|= |A| +|B| -|A∩B|. *The intersection of sets is the upside down ∩ *The union (∪) of two sets is the set of all the elements that are elements of either or both sets Substitute the numbers given in the question: |A∪B|= 7 + 5 - 3 =9
Lee likes both country and pop music. Her playlist has a total of 60 songs that are categorized as pop, country or rock. If a song is listed as both pop and country, it is considered crossover music. If 24 of Lee's songs are classified as rock music only, 30 are pop, and 18 are country, how many are crossover?
The formula for overlapping sets is Total=Group A +Group B-Both +Neither Since the question defines crossover as country and pop, the rock songs can be considered "neither." Plug the given values into the equation: 60= 30 + 18 - crossover + 24 Crossover =12
A bag contains only 4 orange marbles and 2 blue marbles. Latisha wants to get a blue marble from the bag, but she cannot see what color marble she draws until she takes it out of the bag. Latisha will stop drawing marbles as soon as she gets a blue one. If Latisha does not draw a blue marble in 3 attempts, she stops. What is the probability that she will draw a blue marble?
The most efficient way to approach this question is to determine the probability that Latasha will not draw a blue marble in 3 attempts and subtract that from 1 to get the probability that she will. The probability that Latisha will not draw a blue marble on the first attempt is 4/6 -- 2/3 Second attempt 3/5 Third attempt 2/4 or 1/2 So 2/3(3/5)(1/2) = 1/5 1-1/5= 4/5
Pablo is allowed to choose 1 of 3 different fruit beverages and 2 of 4 different healthy grain bars for his afternoon snack. How many different combinations does he have from which to choose?
The number of options Pablo has for the beverage is simply 3, because he can only select one item of the 3 that are available to him. On the other had, the number of options for grain bars needs the combination formula 4! ---- 2! (4-2)! 4X3X2X1 --------- 2X1X2X1 12/2 6 6X3= 18
Restaurant A has 5 appetizers, 20 main courses, and 4 desserts. If a meal consists of 1 appetizer, 1 main course, and 1 dessert, how many different meals can be ordered from restaurant A?
The number of possible outcomes from each set is the number of items in the set. So there are 5 possible appetizers, 20 possible main courses, and 4 possible desserts. 5x20x4= 400 different meals that can be ordered
A 6 sided cube has faces numbered 1 through 6. If the cube is rolled twice, what is the probability that the sum of the two rolls is 8?
The probability of any event equals the number of ways to get the desired outcome divided by the total number of ways for the event to happen. Starting with the denominator, use the fundamental counting principle to compute the total number of ways to roll a cube twice. There are 6 possibilities. (1,2,3,4,5,6) for the first roll and 6 for the second. Giving a total of (6)(6)=36 possibilities for the two rolls. For the numerator, determine the number of possible combinations that will sum to 8. For example, rolling a 2 the first time and a 6 the second time. The full set of options is (2,6) (3,5)(4,4)(5,3) and (6,2). Thus, there are 5 possible combinations that sum to 8, yielding a probability of 5/36.
A bag contains 10 marbles, 4 of which are blue and 6 of which are red. If 2 marbles are removed without replacement, what is the probability that both marbles removed are red?
The probability that the first marble removed will be red is 6/10 = 3/5 The probability that the second marble removed will be red will not be the same, however. There will be one fewer red marble (Note that since we are asking about the odds of picking two red marbles, we are only interested in choosing a second marble if the first was red. Don't concern yourself with situations in which a blue marble is chosen first) If the first marble is red, the probability that the second marble removed will also be red is 5/9 So (3/5) (5/9) =15/45 =1/3
The probability of rain in Greg's town on Tuesday is 0.3. The probability that Greg's teacher will give him a pop quiz on Tuesday is 0.2. the events occur independently of each other. Quantity A: The probability that either or both events occur Quantity B: The probability that neither event occurs C D
The problem indicates that the events occur independently of each other. Therefore, in calculating Quantity A, do not just add both events, even though it is an "or" situation. This is because the probability that both events occur is counted twice. (only add probabilities in an "or" situation when the probabilities are mutually exclusive) Quantity A: 0.3+0.2-(0.3)(0.2)=0.44 Quantity B: Multiply the probability that rain does not occur (0.7) and the probability that the pop quiz does not occur (0.8) (0.7)(0.8) = 0.56
What does it mean when two sets are mutually exclusive
The sets have no common elements
A certain city has 1/3 chance of rain occurring on any given day. In any given 3-day period, what is the probability that the city experiences rain?
The trick here is to calculate the probability that there will not be rain. 2/3 2/3(2/3)(2/3) = 8/27 1-8/27 = 19/27
A group of 12 people who have never met are in a classroom. How many handshakes are exchanged if each person shakes hands exactly once with each of the other people in the room?
There are 11 other people in the room. So, the first person needs to shake hands 11 times. Now, move to the second person: he has already shaken one hand, leaving him 10 others to shake. The third person will need to shake hands with 9 people, and so on 11+10+9+8+7+6+5+4+3+2+1
What is the probability of the result of 4 independent coin flips being exactly 1 head and 3 tails?
There are two possible outcomes for each flip of the coin: heads or tails. You might be tempted to think that the total number of possible outcomes for 4 consecutive flips would be 4x2=8, but remember that the coin is flipped once and then a second time and then a third time and then a fourth time, so the total number of possible outcomes is actually 2⁴ = 16 If only one head is the result, that could occur on any one of the four flips, so there are 4 desired outcomes. The probability is thus 4/16 =1/4
A number cub has six faces numbered 1 through 6. If the cube is rolled twice, what is the probability that at least one of the rolls will result in a number greater than 4?
This is an "at least" solution. So, you have to subtract 1-x meaning: 4/6 can be reduced to 2/3 2/3(2/3) = 4/9 1-4/9= 5/9
In a school of 150 students, 75 study Latin, 110 study Spanish, and 11 study neither. Quantity A: The number of students who study only Latin Quantity B: 46 Quantity C: Quantity D:
This is the probability/union problem: |A ∪ B| = |A| +|B| - |A ∪ B| 139= 75 + 110 - |A ∪ B| 139=185 185-139=46 75-46= 29 Quantity B is greater than 29 (Quantity A)
How do you find the number of elements in a set that is mutually exclusive?
This means mutually exclusive: |A ∪ C|=∅ The equation is then: |A ∪ C|= |A| + |C|
BurgerTown offers many options for customizing a burger. There are 3 types of meats and 7 condiments: lettuce, tomatoes, pickles, onions, ketchup, mustard and a special sauce. A burger must include meat, but may include as many or as few condiments as the customer wants. How many different burgers are possible? A) 8! B) (3)(7!) C) (3)(8!) D) (8)(2⁷) E) (3)(2⁷)
This problem relies on the fundamental counting principle, which states that the total number of choices is equal to the product of the independent choices. The key to this problem is realizing how many choices there are for each option. For meat, there are 3 choices. For each of the condiments there are exactly 2 choices: yes or no. The only real choice regarding each condiment is whether to include it at all. As there are 7 condiments, the total number of choices is: 3(2)(2)(2)(2)(2)(2)(2) E
What is a permutation?
Within any group of items or people, there are multiple arrangements or permutations Ie: A,B,C (ABC, ACB, BAC, BCA, CAB, CBA)
What is the probability of tossing a fair coin four consecutive times and having the coin land heads up exactly once?
You know that all possible outcomes of the experiment are equally likely to occur, so this is the equation you will use Probability= number of desired outcomes/number of possible outcomes number of possible outcomes: Since a coin is tossed four times and each toss has two possible outcomes, the total number of outcomes is 2⁴ =16 The number of desired outcomes is 4 So 4/16 =1/4
What is the combination formula?
n!/k! (n-k)! n=number of items as a whole k= number of items in each subgroup !=factoral=> (5)(4)(3)(2)(1)
