Q: Suppose events $$E$$ and $$F$$ are such that $$P(E \ \text{and} \ F) = 0.7$$ and $$P(E) = 0.8. $$What is $$P(F|E)$$?

$$P(F|E) = \dfrac{P(E \ \text{and} \ F)}{P(E)} = \dfrac{0.7}{0.8} = 0.875$$

Q: If $$P(E) = 0.6$$ and $$P(F|E) = 0.3$$, what is $$P(E \ \text{and} \ F)$$?

$$P(E \ \text{and} \ F) = P(E) \times P(F|E) = 0.6 \times 0.3 = 0.18$$

Q: In a study, 25% of marriages involve a woman with a bachelor's degree and last at least 20 years. Census data shows 26% of women have a bachelor's degree. What is the probability that a randomly selected marriage will last at least 20 years, given the woman has a bachelor's degree?

$$P(20+ \ \text{years} | \ \text{bachelor's}) = \dfrac{0.25}{0.26} \approx 0.962$$

Q: A bag contains 10 red, 9 yellow, and 10 purple tulip bulbs (29 total). Two bulbs are drawn without replacement. What is the probability both bulbs are red?

$$P(\text{both red}) = \dfrac{10}{29} \times \dfrac{9}{28} = \dfrac{90}{812} \approx 0.111$$

Q: Using the same bag of tulip bulbs (10 red, 9 yellow, 10 purple), what is the probability that the first bulb is red and the second is yellow (without replacement)?

$$P(\text{red then yellow}) = \dfrac{10}{29} \times \dfrac{9}{28} = \dfrac{90}{812} \approx 0.111$$

Q: A professor randomly selects two students from a class of 10 (Jinita, Dave, Mike, Al, William, Pam, Allison, Neta, Jim, Kristin). What is the probability that Al is chosen first and Jinita second?

$$P(\text{Al first, Jinita second}) = \dfrac{1}{10} \times \dfrac{1}{9} = \dfrac{1}{90} \approx 0.011$$

Q: Given the fatal crash data, among fatal crashes in normal weather, what is the probability that a randomly selected fatal crash occurs during daylight?

$$P(\text{Daylight} | \text{Normal}) = \dfrac{14,307}{14,307 + 5,875 + 8,151 + 1,183 + 65} = \dfrac{14,307}{29,581} \approx 0.484$$

Q: Among fatal crashes that occurred during daylight, what is the probability that the weather was normal?

$$P(\text{Normal} | \text{Daylight}) = \dfrac{14,307}{14,307 + 875 + 219 + 125 + 810} = \dfrac{14,307}{16,336} \approx 0.876$$

Q: Based on the fatal crash data, is 'dark, but lighted' more dangerous in normal weather or in snow/sleet? Use probabilities to justify your answer.

$$P(\text{Dark, Lighted} | \text{Normal}) = \dfrac{8,151}{29,581} \approx 0.276$$; $$P(\text{Dark, Lighted} | \text{Snow/Sleet}) = \dfrac{156}{445} \approx 0.351. $$Thus, 'dark, but lighted' is more dangerous in snow/sleet.

Q: Suppose you draw two bulbs without replacement from the bag (10 red, 9 yellow, 10 purple). What is the probability that the first is yellow and the second is purple?

$$P(\text{yellow then purple}) = \dfrac{9}{29} \times \dfrac{10}{28} = \dfrac{90}{812} \approx 0.111$$

Question 1

Suppose events $$E$$ and $$F$$ are such that $$P(E \ 	ext{and} \ F) = 0.7$$ and $$P(E) = 0.8. $$What is $$P(F|E)$$?

Accepted Answer

$$P(F|E) = \dfrac{P(E \ 	ext{and} \ F)}{P(E)} = \dfrac{0.7}{0.8} = 0.875$$

Question 2

If $$P(E) = 0.6$$ and $$P(F|E) = 0.3$$, what is $$P(E \ 	ext{and} \ F)$$?

Accepted Answer

$$P(E \ 	ext{and} \ F) = P(E) 	imes P(F|E) = 0.6 	imes 0.3 = 0.18$$

Question 3

In a study, 25% of marriages involve a woman with a bachelor's degree and last at least 20 years. Census data shows 26% of women have a bachelor's degree. What is the probability that a randomly selected marriage will last at least 20 years, given the woman has a bachelor's degree?

Accepted Answer

$$P(20+ \ 	ext{years} | \ 	ext{bachelor's}) = \dfrac{0.25}{0.26} \approx 0.962$$

Question 4

A bag contains 10 red, 9 yellow, and 10 purple tulip bulbs (29 total). Two bulbs are drawn without replacement. What is the probability both bulbs are red?

Accepted Answer

$$P(	ext{both red}) = \dfrac{10}{29} 	imes \dfrac{9}{28} = \dfrac{90}{812} \approx 0.111$$

Question 5

Using the same bag of tulip bulbs (10 red, 9 yellow, 10 purple), what is the probability that the first bulb is red and the second is yellow (without replacement)?

Accepted Answer

$$P(	ext{red then yellow}) = \dfrac{10}{29} 	imes \dfrac{9}{28} = \dfrac{90}{812} \approx 0.111$$

Question 6

A professor randomly selects two students from a class of 10 (Jinita, Dave, Mike, Al, William, Pam, Allison, Neta, Jim, Kristin). What is the probability that Al is chosen first and Jinita second?

Accepted Answer

$$P(	ext{Al first, Jinita second}) = \dfrac{1}{10} 	imes \dfrac{1}{9} = \dfrac{1}{90} \approx 0.011$$

Question 7

Given the fatal crash data, among fatal crashes in normal weather, what is the probability that a randomly selected fatal crash occurs during daylight?

Accepted Answer

$$P(	ext{Daylight} | 	ext{Normal}) = \dfrac{14,307}{14,307 + 5,875 + 8,151 + 1,183 + 65} = \dfrac{14,307}{29,581} \approx 0.484$$

Question 8

Among fatal crashes that occurred during daylight, what is the probability that the weather was normal?

Accepted Answer

$$P(	ext{Normal} | 	ext{Daylight}) = \dfrac{14,307}{14,307 + 875 + 219 + 125 + 810} = \dfrac{14,307}{16,336} \approx 0.876$$

Question 9

Based on the fatal crash data, is 'dark, but lighted' more dangerous in normal weather or in snow/sleet? Use probabilities to justify your answer.

Accepted Answer

$$P(	ext{Dark, Lighted} | 	ext{Normal}) = \dfrac{8,151}{29,581} \approx 0.276$$; $$P(	ext{Dark, Lighted} | 	ext{Snow/Sleet}) = \dfrac{156}{445} \approx 0.351. $$Thus, 'dark, but lighted' is more dangerous in snow/sleet.

Question 10

Suppose you draw two bulbs without replacement from the bag (10 red, 9 yellow, 10 purple). What is the probability that the first is yellow and the second is purple?

Accepted Answer

$$P(	ext{yellow then purple}) = \dfrac{9}{29} 	imes \dfrac{10}{28} = \dfrac{90}{812} \approx 0.111$$

Question 11

If $$P(A) = 0.4$$ and $$P(B|A) = 0.5$$, what is $$P(A \ 	ext{and} \ B)$$?

Accepted Answer

$$P(A \ 	ext{and} \ B) = P(A) 	imes P(B|A) = 0.4 	imes 0.5 = 0.2$$

Question 12

A researcher finds that 30% of marriages last at least 20 years and 20% of marriages involve a woman with a bachelor's degree. If 10% of marriages both last at least 20 years and involve a woman with a bachelor's degree, what is the probability that a randomly selected marriage involves a woman with a bachelor's degree, given it lasted at least 20 years?

Accepted Answer

$$P(	ext{bachelor's} | 20+ \ 	ext{years}) = \dfrac{0.10}{0.30} = 0.333$$

Question 13

If two bulbs are drawn without replacement from the bag (10 red, 9 yellow, 10 purple), what is the probability that both bulbs are the same color?

Accepted Answer

$$P(	ext{same color}) = \dfrac{10}{29} 	imes \dfrac{9}{28} + \dfrac{9}{29} 	imes \dfrac{8}{28} + \dfrac{10}{29} 	imes \dfrac{9}{28} = \dfrac{90}{812} + \dfrac{72}{812} + \dfrac{90}{812} = \dfrac{252}{812} \approx 0.310$$

Question 14

A fatal crash occurs in 'other' weather conditions. What is the probability it happened at dawn/dusk?

Accepted Answer

$$P(	ext{Dawn/Dusk} | 	ext{Other}) = \dfrac{40}{125 + 54 + 220 + 40 + 9} = \dfrac{40}{448} \approx 0.089$$

Question 15

If $$P(E) = 0.7$$, $$P(F) = 0.5$$, and $$E$$ and $$F$$ are independent, what is $$P(E \ 	ext{and} \ F)$$?

Accepted Answer

$$P(E \ 	ext{and} \ F) = P(E) 	imes P(F) = 0.7 	imes 0.5 = 0.35$$

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