Q: A committee of 4 is to be formed from 7 men and 5 women. How many ways can the committee be formed if it must contain exactly 2 women?

$$\binom{5}{2} \times \binom{7}{2} = 10 \times 21 = 210$$

Q: A box contains 5 red balls and 3 blue balls. If two balls are drawn one after another without replacement, what is the probability that both are red?

$$\frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$$

Q: A game costs $$2 to play. You roll a fair die: if you roll a 6, you win 10$$; if you roll a 4 or 5, you win $4; otherwise, you win nothing. What is the expected value of your net gain per game?

$$(-2) \times \frac{3}{6} + 2 \times \frac{2}{6} + 8 \times \frac{1}{6} = -1 + \frac{4}{6} + \frac{8}{6} = -1 + 2 = 1$$

Q: A sample of 5,000 people is surveyed, and 60% support a new law. What is the margin of error for this survey at the 95% confidence level?

$$\frac{1}{\sqrt{5000}} \times 100\% \approx 1.4\%$$

Question 1

A restaurant offers 8 types of salads, 5 types of soups, and 4 types of desserts. If a customer wants to order one salad, one soup, and one dessert, how many different meal combinations are possible?

Accepted Answer

$$8 	imes 5 	imes 4 = 160$$

Question 2

How many distinct ways can the letters in the word 'BALLOON' be arranged?

Accepted Answer

$$7! / (2!2!) = 1260$$

Question 3

A committee of 4 is to be formed from 7 men and 5 women. How many ways can the committee be formed if it must contain exactly 2 women?

Accepted Answer

$$\binom{5}{2} 	imes \binom{7}{2} = 10 	imes 21 = 210$$

Question 4

If a fair die is rolled, what is the probability of rolling a number greater than 4?

Accepted Answer

$$\frac{2}{6} = \frac{1}{3}$$

Question 5

A survey of 300 people found that 120 like tea, 180 like coffee, and 60 like both. What is the probability that a randomly selected person likes tea or coffee?

Accepted Answer

$$\frac{120}{300} + \frac{180}{300} - \frac{60}{300} = \frac{240}{300} = 0.8$$

Question 6

If the odds in favor of an event are 3 to 7, what is the probability of the event?

Accepted Answer

$$\frac{3}{3+7} = \frac{3}{10}$$

Question 7

A box contains 5 red balls and 3 blue balls. If two balls are drawn one after another without replacement, what is the probability that both are red?

Accepted Answer

$$\frac{5}{8} 	imes \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$$

Question 8

A game costs $$2 to play. You roll a fair die: if you roll a 6, you win 10$$; if you roll a 4 or 5, you win $4; otherwise, you win nothing. What is the expected value of your net gain per game?

Accepted Answer

$$(-2) 	imes \frac{3}{6} + 2 	imes \frac{2}{6} + 8 	imes \frac{1}{6} = -1 + \frac{4}{6} + \frac{8}{6} = -1 + 2 = 1$$

Question 9

A sample of 5,000 people is surveyed, and 60% support a new law. What is the margin of error for this survey at the 95% confidence level?

Accepted Answer

$$\frac{1}{\sqrt{5000}} 	imes 100\% \approx 1.4\%$$

Question 10

The heights of adult women are normally distributed with a mean of 65 inches and a standard deviation of 3 inches. What percentage of women are taller than 68 inches?

Accepted Answer

$$z = \frac{68 - 65}{3} = 1$$;$$P(z \u003e 1) = 100\% - 84.13\% = 15.87\%$$

Question 11

A set of data has the following values: 3, 7, 7, 10, 12, 12, 12, 15. What is the mode of this data set?

Accepted Answer

12

Question 12

Given the data set: 5, 8, 12, 15, 20, what is the midrange?

Accepted Answer

$$\frac{5 + 20}{2} = 12.5$$

Question 13

A class of 10 students has test scores: 60, 62, 65, 68, 70, 72, 75, 78, 80, 85. What is the median score?

Accepted Answer

$$\frac{70 + 72}{2} = 71$$

Question 14

The standard deviation of the data set 2, 4, 6, 8 is closest to:

Accepted Answer

$$\sqrt{\frac{(2-5)^2$ + (4-5)^2$ + (6-5)^2$ + (8-5)^2$}{3}} = \sqrt{\frac{9+1+1+9}{3}} = \sqrt{\frac{20}{3}} \approx 2.58$$

Question 15

A random sample of 400 people is taken from a city of 50,000. Which of the following best describes the sample?

Accepted Answer

A subset of the population

Question 16

If the mean of a data set is 10 and the standard deviation is 2, what is the z-score for a data value of 14?

Accepted Answer

$$z = \frac{14 - 10}{2} = 2$$

Question 17

A standard deck of 52 cards is shuffled. What is the probability of drawing a card that is either a heart or a king?

Accepted Answer

$$P(heart) + P(king) - P(heart \cap king) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$$

Question 18

A frequency distribution of test scores shows the greatest frequency in the 70-79 range. What does this indicate about the data?

Accepted Answer

Most students scored between 70 and 79

Question 19

If the probability of an event is $$0.2$$, what are the odds against the event?

Accepted Answer

$$\frac{0.8}{0.2} = 4:1$$

Probability, Counting Principles, and Descriptive Statistics: Study Guide

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