Q: You deal a hand of three cards, one at a time, from a standard deck. What is the probability that you have at least one face card?

$$1 - \left(\frac{40}{52} \times \frac{39}{51} \times \frac{38}{50}\right)$$

Q: A survey finds that the probability a voter cares about environmental issues is $$0.64$$, and the probability they voted in the last election is $$0.72. If $$the probability that a voter both cares about environmental issues and voted is $$0.48$$, are these events independent?

No, because $$0.64 \times 0.72 = 0.4608 \neq 0.48$$

Q: A group of volunteers is divided into 40 men and 70 women. 13 of the women have high blood pressure, and 8 of the men have high blood pressure. Are high blood pressure and gender independent?

No, because $$P(\text{high BP} | \text{woman}) \neq P(\text{high BP})$$

Q: In a population, 3.5% are infected with a virus. A test for the disease has a 98% sensitivity (true positive rate) and a 97% specificity (true negative rate). What is the probability that a person who tests positive actually has the disease? (Use Bayes' theorem.)

$$\frac{0.035 \times 0.98}{0.035 \times 0.98 + 0.965 \times 0.03}$$

Q: A company interviews 10 job applicants. The lie detector test is 90% accurate for liars and 95% accurate for truth-tellers. If 20% of applicants lie, what is the probability that a randomly selected applicant who fails the test is actually lying?

$$\frac{0.2 \times 0.9}{0.2 \times 0.9 + 0.8 \times 0.05}$$

Q: A table shows the smoking habits of a group of airline staff by gender. If 30% of males and 20% of females are heavy smokers, is heavy smoking independent of gender?

No, because $$P(\text{heavy smoker} | \text{male}) \neq P(\text{heavy smoker})$$

Question 1

A sporting goods store uses a 'Wheel of Savings' with the following probabilities: 10% for a 40% discount, 25% for a 25% discount, 45% for a 10% discount, and 20% for no discount. What is the probability that a customer gets at least a 25% discount?

Accepted Answer

$$0.55$$

Question 2

If three customers spin the 'Wheel of Savings' in a row, what is the probability that all three get a 10% discount? (Assume independence.)

Accepted Answer

$$0.45^3$$

Question 3

A survey reveals that 8% of people are motivated by reading books, 6% by watching TV, and 3% by both. What is the probability that a person is motivated by either books or TV?

Accepted Answer

$$0.08 + 0.06 - 0.03 = 0.11$$

Question 4

In a college, 15% of students speak Spanish, 7% speak French, and 4% speak both. What is the probability that a randomly selected student speaks either Spanish or French?

Accepted Answer

$$0.15 + 0.07 - 0.04 = 0.18$$

Question 5

A table shows that 42% of voters are Democrats, 36% are Republicans, and 22% are Independents. If 60% of Republicans support gun control, what is the probability that a randomly selected voter is a Republican who supports gun control?

Accepted Answer

$$0.36 	imes 0.60 = 0.216$$

Question 6

You deal a hand of three cards, one at a time, from a standard deck. What is the probability that you have at least one face card?

Accepted Answer

$$1 - \left(\frac{40}{52} 	imes \frac{39}{51} 	imes \frac{38}{50}ight)$$

Question 7

A survey finds that the probability a voter cares about environmental issues is $$0.64$$, and the probability they voted in the last election is $$0.72. If $$the probability that a voter both cares about environmental issues and voted is $$0.48$$, are these events independent?

Accepted Answer

No, because $$0.64 	imes 0.72 = 0.4608 
eq 0.48$$

Question 8

A group of volunteers is divided into 40 men and 70 women. 13 of the women have high blood pressure, and 8 of the men have high blood pressure. Are high blood pressure and gender independent?

Accepted Answer

No, because $$P(	ext{high BP} | 	ext{woman}) 
eq P(	ext{high BP})$$

Question 9

In a population, 3.5% are infected with a virus. A test for the disease has a 98% sensitivity (true positive rate) and a 97% specificity (true negative rate). What is the probability that a person who tests positive actually has the disease? (Use Bayes' theorem.)

Accepted Answer

$$\frac{0.035 	imes 0.98}{0.035 	imes 0.98 + 0.965 	imes 0.03}$$

Question 10

A company interviews 10 job applicants. The lie detector test is 90% accurate for liars and 95% accurate for truth-tellers. If 20% of applicants lie, what is the probability that a randomly selected applicant who fails the test is actually lying?

Accepted Answer

$$\frac{0.2 	imes 0.9}{0.2 	imes 0.9 + 0.8 	imes 0.05}$$

Question 11

A box contains 12 batteries, of which 4 are still working. If you randomly select 3 batteries without replacement, what is the probability that at least one is working?

Accepted Answer

$$1 - \frac{\binom{8}{3}}{\binom{12}{3}}$$

Question 12

A table shows the smoking habits of a group of airline staff by gender. If 30% of males and 20% of females are heavy smokers, is heavy smoking independent of gender?

Accepted Answer

No, because $$P(	ext{heavy smoker} | 	ext{male}) 
eq P(	ext{heavy smoker})$$

Question 13

Suppose you are waiting in line and observe two gold winners in a row on the 'Wheel of Savings.' If the probability of winning gold is $$0.10$$, what is the probability that the next customer also wins gold?

Accepted Answer

$$0.10$$

Question 14

If the probability that a student passes Statistics is $$0.85$$ and the probability that a student passes Calculus is $$0.75$$, and the probability that a student passes both is $$0.70$$, what is the probability that a student passes at least one of the two courses?

Accepted Answer

$$0.85 + 0.75 - 0.70 = 0.90$$

Probability and Independence: Study Notes for Statistics

Guida di studio - Domande di pratica

Study guide - Flashcard