Q: A fair six-sided die is rolled twice. What is the probability that both rolls result in a 2?

$$P(2 \text{ on first roll}) \times P(2 \text{ on second roll}) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$$

Q: Which of the following statements best describes independent events?

Two events A and B are independent if $$P(A \cap B) = P(A) \times P(B)$$

Q: A contingency table shows that 30 students are smokers and 70 are non-smokers. If 20 of the smokers are also part-timers, and 40 of the non-smokers are part-timers, what is the probability that a randomly selected student is a part-timer?

$$P(\text{part-timer}) = \frac{20 + 40}{30 + 70} = \frac{60}{100} = 0.6$$

Q: Given $$P( ext{death}| ext{no water}) = 0.9$$ and $$P( ext{death}| ext{water}) = 0.2$$, if the probability that your friend forgets to water is 0.3, what is the overall probability that the plant dies?

$$P( ext{death}) = P( ext{death}| ext{no water}) \times P( ext{no water}) + P( ext{death}| ext{water}) \times P( ext{water}) = 0.9 \times 0.3 + 0.2 \times 0.7 = 0.27 + 0.14 = 0.41$$

Q: A fair coin is tossed four times. What is the probability of getting all heads?

$$P(\text{all heads}) = \left(\frac{1}{2}\$$right)^4$$ = \frac{1}{16}$$

Q: A golf club has 600 members, 400 of whom use the tennis courts regularly. If a member is selected at random, what is the probability that they use the tennis courts?

$$P(\text{uses tennis courts}) = \frac{400}{600} = \frac{2}{3}$$

Question 1

Suppose you have two events, A and B, where $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.2. $$What is the conditional probability $$P(A|B)$$?

Accepted Answer

$$P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.2}{0.5} = 0.4$$

Question 2

A fair six-sided die is rolled twice. What is the probability that both rolls result in a 2?

Accepted Answer

$$P(2 	ext{ on first roll}) 	imes P(2 	ext{ on second roll}) = \frac{1}{6} 	imes \frac{1}{6} = \frac{1}{36}$$

Question 3

Which of the following statements best describes independent events?

Accepted Answer

Two events A and B are independent if $$P(A \cap B) = P(A) 	imes P(B)$$

Question 4

A contingency table shows that 30 students are smokers and 70 are non-smokers. If 20 of the smokers are also part-timers, and 40 of the non-smokers are part-timers, what is the probability that a randomly selected student is a part-timer?

Accepted Answer

$$P(	ext{part-timer}) = \frac{20 + 40}{30 + 70} = \frac{60}{100} = 0.6$$

Question 5

Given $$P(	ext{death}|	ext{no water}) = 0.9$$ and $$P(	ext{death}|	ext{water}) = 0.2$$, if the probability that your friend forgets to water is 0.3, what is the overall probability that the plant dies?

Accepted Answer

$$P(	ext{death}) = P(	ext{death}|	ext{no water}) 	imes P(	ext{no water}) + P(	ext{death}|	ext{water}) 	imes P(	ext{water}) = 0.9 	imes 0.3 + 0.2 	imes 0.7 = 0.27 + 0.14 = 0.41$$

Question 6

A standard deck of 52 cards is shuffled. What is the probability of drawing a jack or a heart?

Accepted Answer

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

Question 7

If the mean of a data set is 20 and the standard deviation is 5, what is the z-score of a value 35?

Accepted Answer

$$z = \frac{35 - 20}{5} = 3$$

Question 8

A boxplot of assignment completion times shows a median of 40 minutes, a lower quartile of 30 minutes, and an upper quartile of 62 minutes. What is the interquartile range (IQR)?

Accepted Answer

$$IQR = Q_3$ - Q_1$ = 62 - 30 = 32$$

Question 9

If the distribution of ages in a population is right-skewed, which of the following is true?

Accepted Answer

The mean is greater than the median.

Question 10

A fair coin is tossed four times. What is the probability of getting all heads?

Accepted Answer

$$P(	ext{all heads}) = \left(\frac{1}{2}\$$right)^4$$ = \frac{1}{16}$$

Question 11

A golf club has 600 members, 400 of whom use the tennis courts regularly. If a member is selected at random, what is the probability that they use the tennis courts?

Accepted Answer

$$P(	ext{uses tennis courts}) = \frac{400}{600} = \frac{2}{3}$$

Question 12

If $$P(A) = 0.7$$ and $$P(B) = 0.5$$, and A and B are independent, what is $$P(A \cap B)$$?

Accepted Answer

$$P(A \cap B) = P(A) 	imes P(B) = 0.7 	imes 0.5 = 0.35$$

Question 13

A contingency table shows 15 males and 25 females. 10 males and 20 females are weight watchers. What is the probability that a randomly selected person is a female and a weight watcher?

Accepted Answer

$$P(	ext{female and WW}) = \frac{20}{15 + 25} = \frac{20}{40} = 0.5$$

Question 14

If $$P(A|B) = 0.3$$ and $$P(B) = 0.5$$, what is $$P(A \cap B)$$?

Accepted Answer

$$P(A \cap B) = P(A|B) 	imes P(B) = 0.3 	imes 0.5 = 0.15$$

Question 15

A plant has a 0.3 probability of not being watered and a 0.9 probability of dying if not watered. What is the joint probability that the plant is not watered and dies?

Accepted Answer

$$P(	ext{no water and death}) = P(	ext{no water}) 	imes P(	ext{death} | 	ext{no water}) = 0.3 	imes 0.9 = 0.27$$

Question 16

If the mean of a data set is 32, the median is 40, and the mode is 16, what can you say about the shape of the distribution?

Accepted Answer

The distribution is skewed right.

Question 17

A committee of 4 is to be selected from 9 people. How many different committees are possible?

Accepted Answer

$$\binom{9}{4} = \frac{9!}{4!5!} = 126$$

Question 18

If $$P(A) = 0.6$$, $$P(B) = 0.3$$, and $$P(A \cap B) = 0.18$$, are A and B independent?

Accepted Answer

Yes, because $$P(A \cap B) = P(A) 	imes P(B) = 0.6 	imes 0.3 = 0.18$$

Probability and Descriptive Statistics: Study Guide

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