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Probability and Counting Principles Quiz

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If you roll a fair six-sided die, what is the probability of rolling a prime number?

1/3

Explanation

Prime numbers on a six-sided die: 2, 3, 5. Probability = Number of prime outcomes / Total outcomes = 3/6 = 1/3.

In a group of 10 people, how many ways can you choose a president and a vice president?

Explanation

Number of ways to choose president and vice president from 10 people = 10P2 = 10! / (10-2)! = 10 * 9 = 90 / 2 = 45.

In a group of 6 students, how many different committees of 3 can be formed?

Explanation

Number of ways to choose a committee of 3 from 6 students = 6C3 = 20 / 2 = 15.

If you roll two fair six-sided dice, what is the probability that the sum of the numbers rolled is 7?

1/6

Explanation

Number of ways to get a sum of 7 = 6 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1). Total outcomes = 6 * 6 = 36. Probability = 6/36 = 1/6.

In a class of 25 students, how many different pairs of students can be formed for a dance competition?

600

Explanation

Number of ways to form pairs from 25 students = 25C2 = 25! / (2! * (25-2)!) = 300 / 1 = 600.

You have 5 different types of sandwiches and want to choose 2 to pack for lunch. In how many ways can you choose your lunch?

Explanation

Number of ways to choose 2 sandwiches from 5 = 5C2 = 10 / 2 = 20.

If you flip a fair coin three times, what is the probability of getting exactly two heads?

3/8

Explanation

Probability of getting exactly two heads in three coin flips: HHT, HTH, THH. Probability = 3/2^3 = 3/8.

A bag contains 4 red balls, 3 blue balls, and 5 green balls. If a ball is randomly chosen from the bag, what is the probability of choosing a blue ball?

3/12

Explanation

Probability of choosing a blue ball = Number of blue balls / Total balls = 3 / (4+3+5) = 3/12.

If you draw two cards from a standard deck of 52 cards without replacement, what is the probability that both cards are hearts?

1/169

Explanation

Probability of drawing two hearts = (Number of hearts / Total cards) * ((Number of hearts - 1) / (Total cards - 1)) = (13/52) * (12/51) = 1/169.

#10

A jar contains 5 red balls, 3 blue balls, and 2 green balls. If you randomly draw two balls without replacement, what is the probability of drawing one red ball and one blue ball?

15/40

Explanation

Probability of drawing one red and one blue ball = (Number of ways to choose 1 red and 1 blue / Total ways to draw 2 balls) = (5C1 * 3C1) / 10C2 = 15/40.

#11

You have 4 different pairs of shoes. In how many ways can you choose 2 shoes to wear if each shoe must be from a different pair?

Explanation

Number of ways to choose 2 shoes from different pairs = 4C1 * 2 = 4 * 2 = 8. Each pair has 2 choices, and there are 4 pairs.

#12

A box contains 5 red balls, 4 blue balls, and 3 green balls. If you draw one ball at random, what is the probability that it is not green?

9/12

Explanation

Probability of not drawing a green ball = (Number of non-green balls / Total balls) = (5+4) / (5+4+3) = 9/12.

#13

In a standard deck of 52 cards, how many ways can you draw two cards of the same suit?

104

Explanation

Number of ways to draw two cards of the same suit = 4 suits * 13C2 = 4 * 78 = 312 / 3 (due to double counting) = 104.

#14

You have 8 different books and want to arrange them on a shelf. In how many ways can you arrange them if two particular books must always be together?

2880

Explanation

Number of ways to arrange 8 books with two specific books together = 7! * 2! = 5040 / 2 = 2880.

#15

A committee of 5 people is to be chosen from a group of 10 men and 8 women. If the committee must consist of 3 men and 2 women, how many different committees are possible?

1260

Explanation

Number of ways to choose 3 men from 10 men = 10C3. Number of ways to choose 2 women from 8 women = 8C2. Total committees = (10C3) * (8C2) = 1260.

#16

You have 8 different colored socks in a drawer. In how many ways can you choose 2 socks to wear if you want to wear one sock of each color?

Explanation

Number of ways to choose 1 sock of each color = 8C1 * 8C1 = 8 * 8 = 64 / 2 (due to identical pairs) = 32. Order matters, so multiply by 2.

#17

If you randomly pick a day of the week and a month of the year, what is the probability that the day is Friday and the month is December?

1/84

Explanation

Probability of picking Friday and December = (1/7) * (1/12) = 1/84.

#18

A bag contains 6 red balls, 4 blue balls, and 2 green balls. If you draw three balls at random without replacement, what is the probability of getting at least one blue ball?

9/11

Explanation

Probability of drawing at least one blue ball = 1 - Probability of drawing no blue balls = 1 - (6C3 / 12C3) = 9/11.

#19

A committee of 4 people is to be chosen from a group of 7 men and 5 women. If the committee must consist of 2 men and 2 women, how many different committees are possible?

630

Explanation

Number of ways to choose 2 men from 7 men = 7C2. Number of ways to choose 2 women from 5 women = 5C2. Total committees = (7C2) * (5C2) = 630.

#20

A bag contains 7 red balls and 5 blue balls. If you draw two balls at random without replacement, what is the probability that both balls are blue?

1/24

Explanation

Probability of drawing two blue balls = (Number of blue balls / Total balls) * ((Number of blue balls - 1) / (Total balls - 1)) = (5/12) * (4/11) = 1/24.

Probability and Counting Principles Quiz

If you roll a fair six-sided die, what is the probability of rolling a prime number?

In a group of 10 people, how many ways can you choose a president and a vice president?

In a group of 6 students, how many different committees of 3 can be formed?

If you roll two fair six-sided dice, what is the probability that the sum of the numbers rolled is 7?

In a class of 25 students, how many different pairs of students can be formed for a dance competition?

You have 5 different types of sandwiches and want to choose 2 to pack for lunch. In how many ways can you choose your lunch?

If you flip a fair coin three times, what is the probability of getting exactly two heads?

A bag contains 4 red balls, 3 blue balls, and 5 green balls. If a ball is randomly chosen from the bag, what is the probability of choosing a blue ball?

If you draw two cards from a standard deck of 52 cards without replacement, what is the probability that both cards are hearts?

A jar contains 5 red balls, 3 blue balls, and 2 green balls. If you randomly draw two balls without replacement, what is the probability of drawing one red ball and one blue ball?

You have 4 different pairs of shoes. In how many ways can you choose 2 shoes to wear if each shoe must be from a different pair?

A box contains 5 red balls, 4 blue balls, and 3 green balls. If you draw one ball at random, what is the probability that it is not green?

In a standard deck of 52 cards, how many ways can you draw two cards of the same suit?

You have 8 different books and want to arrange them on a shelf. In how many ways can you arrange them if two particular books must always be together?

A committee of 5 people is to be chosen from a group of 10 men and 8 women. If the committee must consist of 3 men and 2 women, how many different committees are possible?

You have 8 different colored socks in a drawer. In how many ways can you choose 2 socks to wear if you want to wear one sock of each color?

If you randomly pick a day of the week and a month of the year, what is the probability that the day is Friday and the month is December?

A bag contains 6 red balls, 4 blue balls, and 2 green balls. If you draw three balls at random without replacement, what is the probability of getting at least one blue ball?

A committee of 4 people is to be chosen from a group of 7 men and 5 women. If the committee must consist of 2 men and 2 women, how many different committees are possible?

A bag contains 7 red balls and 5 blue balls. If you draw two balls at random without replacement, what is the probability that both balls are blue?

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