Q: A researcher claims a new method increases the likelihood of a baby being a girl. In a test of 20 births, all 20 are girls. Using the classical approach, what is the probability of this outcome if the method has no effect (i.e., probability of girl is 0.5 for each birth)?

$$P(20\ \text{girls}) = (0.5$$)^{20}$$ = 0.000000954$$

Q: In a Pew Research Center survey, 366 adults answered 'yes' and 1637 answered 'no' to seeing a ghost. What is the probability that a randomly selected adult answered 'yes'?

$$P(\text{yes}) = \dfrac{366}{366 + 1637} = 0.182$$

Q: A roulette wheel has 38 slots (numbers 1-36, 0, and 00). What is the probability of winning a bet on an even number?

$$P(\text{even}) = \dfrac{18}{38} = 0.474$$

Q: If the probability of a household having a smartphone is $$0.87$$, what is the probability that a randomly selected household does NOT have a smartphone?

$$P(\text{no smartphone}) = 1 - 0.87 = 0.13$$

Q: A deck of cards contains 52 cards. What is the probability of drawing either a red club or a black diamond in a single draw?

$$P(\text{red club or black diamond}) = 0$$

Q: If the probability of an airline crash is $$\dfrac{16}{39,000,000}$$, what are the actual odds against an airline crash?

$$\text{Odds against} = \dfrac{P(\text{no crash})}{P(\text{crash})} = \dfrac{38,999,984}{16} = 2,437,499:1$$

Q: A total of 50 adults were drug tested: 45 positive, 5 negative. If two are selected with replacement, what is the probability both test positive?

$$P(\text{both positive}) = \dfrac{45}{50} \times \dfrac{45}{50} = 0.81$$

Q: In a group of high school drivers, 731 texted while driving and drank alcohol, 3054 texted but did not drink, 156 did not text but drank, and 4564 did neither. What is the probability that a randomly selected driver either texted while driving or drank alcohol?

$$P(\text{text or drink}) = \dfrac{731 + 3054 + 156}{8505} + \dfrac{731 + 156}{8505} - \dfrac{731}{8505} = \dfrac{3941 + 887 - 731}{8505} = \dfrac{4097}{8505} = 0.482$$

Q: If the probability of a hard drive failing in a year is $$0.0366$$, what is the probability that two independent drives both fail within a year?

$$P(\text{both fail}) = 0.0366 \times 0.0366 = 0.00134$$

Q: If $$P(A) = 0.3$$ and $$P(B) = 0.5$$, and A and B are independent, what is $$P(A \text{ and } B)$$?

$$P(A \text{ and } B) = P(A) \times P(B) = 0.3 \times 0.5 = 0.15$$

Question 1

A researcher claims a new method increases the likelihood of a baby being a girl. In a test of 20 births, all 20 are girls. Using the classical approach, what is the probability of this outcome if the method has no effect (i.e., probability of girl is 0.5 for each birth)?

Accepted Answer

$$P(20\ 	ext{girls}) = (0.5$$)^{20}$$ = 0.000000954$$

Question 2

In a Pew Research Center survey, 366 adults answered 'yes' and 1637 answered 'no' to seeing a ghost. What is the probability that a randomly selected adult answered 'yes'?

Accepted Answer

$$P(	ext{yes}) = \dfrac{366}{366 + 1637} = 0.182$$

Question 3

A roulette wheel has 38 slots (numbers 1-36, 0, and 00). What is the probability of winning a bet on an even number?

Accepted Answer

$$P(	ext{even}) = \dfrac{18}{38} = 0.474$$

Question 4

If the probability of a household having a smartphone is $$0.87$$, what is the probability that a randomly selected household does NOT have a smartphone?

Accepted Answer

$$P(	ext{no smartphone}) = 1 - 0.87 = 0.13$$

Question 5

A deck of cards contains 52 cards. What is the probability of drawing either a red club or a black diamond in a single draw?

Accepted Answer

$$P(	ext{red club or black diamond}) = 0$$

Question 6

If the probability of an airline crash is $$\dfrac{16}{39,000,000}$$, what are the actual odds against an airline crash?

Accepted Answer

$$	ext{Odds against} = \dfrac{P(	ext{no crash})}{P(	ext{crash})} = \dfrac{38,999,984}{16} = 2,437,499:1$$

Question 7

A total of 50 adults were drug tested: 45 positive, 5 negative. If two are selected with replacement, what is the probability both test positive?

Accepted Answer

$$P(	ext{both positive}) = \dfrac{45}{50} 	imes \dfrac{45}{50} = 0.81$$

Question 8

In a group of high school drivers, 731 texted while driving and drank alcohol, 3054 texted but did not drink, 156 did not text but drank, and 4564 did neither. What is the probability that a randomly selected driver either texted while driving or drank alcohol?

Accepted Answer

$$P(	ext{text or drink}) = \dfrac{731 + 3054 + 156}{8505} + \dfrac{731 + 156}{8505} - \dfrac{731}{8505} = \dfrac{3941 + 887 - 731}{8505} = \dfrac{4097}{8505} = 0.482$$

Question 9

If the probability of a hard drive failing in a year is $$0.0366$$, what is the probability that two independent drives both fail within a year?

Accepted Answer

$$P(	ext{both fail}) = 0.0366 	imes 0.0366 = 0.00134$$

Question 10

If $$P(A) = 0.3$$ and $$P(B) = 0.5$$, and A and B are independent, what is $$P(A 	ext{ and } B)$$?

Accepted Answer

$$P(A 	ext{ and } B) = P(A) 	imes P(B) = 0.3 	imes 0.5 = 0.15$$

Question 11

A bet on a single number in roulette pays 35:1. If you bet $5 and win, what is your net profit?

Accepted Answer

$$	ext{Net profit} = 35 	imes 5 = 175$$

Question 12

If the probability of an event is $$0.04$$, what is the probability of its complement?

Accepted Answer

$$P(	ext{complement}) = 1 - 0.04 = 0.96$$

Question 13

If you randomly select three employees from a population of 130,639,273, and the probability of a positive drug test is $$0.042$$, what is the probability all three test positive (treating as independent)?

Accepted Answer

$$P(	ext{all positive}) = 0.042^3$ = 0.000074$$

Question 14

Which of the following best describes a simple event?

Accepted Answer

A single outcome that cannot be broken down into simpler components.

Question 15

If $$P(A) = 0.2$$ and $$P(B) = 0.3$$, and A and B are disjoint, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$P(A 	ext{ or } B) = P(A) + P(B) = 0.2 + 0.3 = 0.5$$

Question 16

A sample space consists of all possible simple events. For three births, which of the following is NOT a simple event?

Accepted Answer

Having 2 girls and 1 boy (regardless of order)

Question 17

If you select two adults without replacement from a group of 50 (45 positive, 5 negative drug tests), what is the probability both test positive?

Accepted Answer

$$P(	ext{both positive}) = \dfrac{45}{50} 	imes \dfrac{44}{49} = 0.81$$

Question 18

Which of the following statements about probability is TRUE?

Accepted Answer

$$0 \leq P(A) \leq 1$$

Question 19

If the probability of an event is less than or equal to $$0.05$$, how is this event typically described in hypothesis testing?

Accepted Answer

Unlikely or statistically significant

Chapter 4 and 4.1

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