Q: Are the events 'being a girl' and 'wanting to be popular' independent? Use the definition of independence to justify your answer.

No, because $$P(\text{Popular} | \text{Girl}) \neq P(\text{Popular})$$

Q: Dan’s Diner employs three dishwashers: Al (40% of dishes, 1% breakage), Betty (30%, 1%), and Chuck (30%, 3%). If a dish breaks, what is the probability Chuck was washing it?

$$\frac{0.3 \times 0.03}{0.4 \times 0.01 + 0.3 \times 0.01 + 0.3 \times 0.03}$$

Q: A disease has a prevalence of 0.002. The test is 95% sensitive and 92% specific. What is the probability that a person has the disease given a positive test result?

$$\frac{0.002 \times 0.95}{0.002 \times 0.95 + 0.998 \times 0.08}$$

Q: Leah’s first flight leaves on time with probability 0.15. If on time, luggage makes the connection with probability 0.95; if delayed, with probability 0.65. What is the probability her luggage arrives in Denver with her?

$$0.15 \times 0.95 + 0.85 \times 0.65$$

Q: A junk box contains 12 batteries, 5 of which are dead. What is the probability that the first two you pick are both good?

$$\frac{7}{12} \times \frac{6}{11}$$

Q: In a study, 60% of students study for the AP Statistics exam. 88% of those who study get a 5, and 0.9% of those who do not study get a 5. What is the probability that a student who received a 5 actually studied?

$$\frac{0.6 \times 0.88}{0.6 \times 0.88 + 0.4 \times 0.009}$$

Q: A company buys connectors from three suppliers: A (70%, 1% defective), B (20%, 2%), C (10%, 4%). If a connector is defective, what is the probability it came from supplier A?

$$\frac{0.7 \times 0.01}{0.7 \times 0.01 + 0.2 \times 0.02 + 0.1 \times 0.04}$$

Q: A private college reports: 70% of freshmen attended public schools, 65% of public-school freshmen graduate, 85% of other freshmen graduate. What percent of graduates attended public high school?

$$\frac{0.7 \times 0.65}{0.7 \times 0.65 + 0.3 \times 0.85}$$

Question 1

Given the table of elementary school students' goals, what is the probability that a randomly selected student wants to be popular?

Accepted Answer

$$\frac{141}{478}$$

Question 2

Using the same table, what is the probability that a student wants to be popular given that the student is a girl?

Accepted Answer

$$\frac{91}{251}$$

Question 3

What is the probability that a student is a girl given that the student wants to be popular?

Accepted Answer

$$\frac{91}{141}$$

Question 4

Are the events 'being a girl' and 'wanting to be popular' independent? Use the definition of independence to justify your answer.

Accepted Answer

No, because $$P(	ext{Popular} | 	ext{Girl}) 
eq P(	ext{Popular})$$

Question 5

A weatherman has a 0.015 probability of being completely wrong on any given day. If he makes forecasts for 320 days, what is the probability that at least one forecast is incorrect?

Accepted Answer

$$1 - (1 - 0.015$$)^{320}$$

Question 6

Dan’s Diner employs three dishwashers: Al (40% of dishes, 1% breakage), Betty (30%, 1%), and Chuck (30%, 3%). If a dish breaks, what is the probability Chuck was washing it?

Accepted Answer

$$\frac{0.3 	imes 0.03}{0.4 	imes 0.01 + 0.3 	imes 0.01 + 0.3 	imes 0.03}$$

Question 7

A disease has a prevalence of 0.002. The test is 95% sensitive and 92% specific. What is the probability that a person has the disease given a positive test result?

Accepted Answer

$$\frac{0.002 	imes 0.95}{0.002 	imes 0.95 + 0.998 	imes 0.08}$$

Question 8

Leah’s first flight leaves on time with probability 0.15. If on time, luggage makes the connection with probability 0.95; if delayed, with probability 0.65. What is the probability her luggage arrives in Denver with her?

Accepted Answer

$$0.15 	imes 0.95 + 0.85 	imes 0.65$$

Question 9

In a real-estate market, 64% of homes have garages, 21% have pools, and 17% have both. What is the probability that a home has a pool or a garage?

Accepted Answer

$$0.64 + 0.21 - 0.17$$

Question 10

A junk box contains 12 batteries, 5 of which are dead. What is the probability that the first two you pick are both good?

Accepted Answer

$$\frac{7}{12} 	imes \frac{6}{11}$$

Question 11

A champion bowler has a 70% chance of a strike each frame. What is the probability she goes three consecutive frames without a strike?

Accepted Answer

$$(1 - 0.7)^3$

Question 12

In a study, 60% of students study for the AP Statistics exam. 88% of those who study get a 5, and 0.9% of those who do not study get a 5. What is the probability that a student who received a 5 actually studied?

Accepted Answer

$$\frac{0.6 	imes 0.88}{0.6 	imes 0.88 + 0.4 	imes 0.009}$$

Question 13

Auto accidents: 25% involve teenagers, 65% involve weather, and 16% involve both. Given poor weather, what is the probability that an accident involves a teenager?

Accepted Answer

$$\frac{0.16}{0.65}$$

Question 14

A company buys connectors from three suppliers: A (70%, 1% defective), B (20%, 2%), C (10%, 4%). If a connector is defective, what is the probability it came from supplier A?

Accepted Answer

$$\frac{0.7 	imes 0.01}{0.7 	imes 0.01 + 0.2 	imes 0.02 + 0.1 	imes 0.04}$$

Question 15

A private college reports: 70% of freshmen attended public schools, 65% of public-school freshmen graduate, 85% of other freshmen graduate. What percent of graduates attended public high school?

Accepted Answer

$$\frac{0.7 	imes 0.65}{0.7 	imes 0.65 + 0.3 	imes 0.85}$$

Question 16

At a sobriety checkpoint, 10% of drivers have been drinking. Officers make the right decision 73% of the time. What is the probability that a non-drinking driver is detained for further testing?

Accepted Answer

$$1 - 0.73 = 0.27$$

Question 17

A champion bowler has a 70% chance of a strike each frame. What is the probability she makes her first strike in the third frame?

Accepted Answer

$$(1 - 0.7)^2$ 	imes 0.7$$

Introductory Statistics Study Guide: Conditional Probability & Independence

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