Q: A recent college graduate plans to take three actuarial exams. The probability she passes the first is $$0.9$$, the second (given she passed the first) is $$0.8$$, and the third (given she passed the first two) is $$0.7. $$What is the probability she passes all three exams?

$$0.9 \times 0.8 \times 0.7 = 0.504$$

Q: Suppose that in a certain city, $$46\% of $$voters are Independents, $$30\%$$ are Liberals, and $$24\%$$ are Conservatives. In a recent election, $$35\% of $$Independents, $$62\% of $$Liberals, and $$58\% of $$Conservatives voted. What fraction of voters participated in the election?

$$0.46 \times 0.35 + 0.30 \times 0.62 + 0.24 \times 0.58$$

Q: A deck of $$52$$ cards is randomly divided into $$4$$ hands of $$13$$ cards each. What is the probability that each hand has exactly one ace?

$$\frac{4}{52} \times \frac{3}{39} \times \frac{2}{26} \times \frac{1}{13}$$

Q: Suppose that in a certain 'quit smoking' class, $$48\% of $$women and $$37\% of $$men remained nonsmokers for at least one year. If $$62\% of $$the original class was male, what percentage of those attending the success party were women?

$$\frac{0.48 \times 0.38}{0.48 \times 0.38 + 0.37 \times 0.62} \times 100\%$$

Q: In a medical study, $$32\% of $$women are smokers. An ectopic pregnancy is twice as likely in smokers as in nonsmokers. What percentage of women having ectopic pregnancies are smokers?

$$\frac{2 \times 0.32}{2 \times 0.32 + 0.68} \times 100\%$$

Question 1

Suppose a new car battery has a probability of $$0.8 to $$function for over $$10,000$$ miles, $$0.4$$ for over $$20,000$$ miles, and $$0.1$$ for over $$30,000$$ miles. If the battery is still working after $$10,000$$ miles, what is the probability that its total life will exceed $$20,000$$ miles?

Accepted Answer

$$0.4$$

Question 2

An urn contains $$6$$ white and $$9$$ black balls. If $$4$$ balls are randomly selected without replacement, what is the probability that the first $$2$$ selected are white and the last $$2$$ are black?

Accepted Answer

$$\frac{6}{15} \cdot \frac{5}{14} \cdot \frac{9}{13} \cdot \frac{8}{12}$$

Question 3

A recent college graduate plans to take three actuarial exams. The probability she passes the first is $$0.9$$, the second (given she passed the first) is $$0.8$$, and the third (given she passed the first two) is $$0.7. $$What is the probability she passes all three exams?

Accepted Answer

$$0.9 	imes 0.8 	imes 0.7 = 0.504$$

Question 4

Suppose that in a certain city, $$46\% of $$voters are Independents, $$30\%$$ are Liberals, and $$24\%$$ are Conservatives. In a recent election, $$35\% of $$Independents, $$62\% of $$Liberals, and $$58\% of $$Conservatives voted. What fraction of voters participated in the election?

Accepted Answer

$$0.46 	imes 0.35 + 0.30 	imes 0.62 + 0.24 	imes 0.58$$

Question 5

A deck of $$52$$ cards is randomly divided into $$4$$ hands of $$13$$ cards each. What is the probability that each hand has exactly one ace?

Accepted Answer

$$\frac{4}{52} 	imes \frac{3}{39} 	imes \frac{2}{26} 	imes \frac{1}{13}$$

Question 6

Suppose that in a certain 'quit smoking' class, $$48\% of $$women and $$37\% of $$men remained nonsmokers for at least one year. If $$62\% of $$the original class was male, what percentage of those attending the success party were women?

Accepted Answer

$$\frac{0.48 	imes 0.38}{0.48 	imes 0.38 + 0.37 	imes 0.62} 	imes 100\%$$

Question 7

A random variable $$X$$ has probability mass function $$p(i) = c \lambda^i / i!$$ for $$i = 0, 1, 2, ...$$, where $$\lambda \u003e 0. $$What is the value of $$c$$?

Accepted Answer

$$c = e$$^{-\lambda}$$

Question 8

If $$X is a $$binomial random variable with parameters $$n$$ and $$p$$, what is its expected value $$E[X]$$?

Accepted Answer

$$np$$

Question 9

Suppose a Poisson random variable $$X$$ has parameter $$\lambda. $$What is the variance $$Var(X)$$?

Accepted Answer

$$\lambda$$

Question 10

A geometric random variable $$X$$ with parameter $$p$$ represents the number of trials until the first success. What is $$E[X]$$?

Accepted Answer

$$\frac{1}{p}$$

Question 11

In a medical study, $$32\% of $$women are smokers. An ectopic pregnancy is twice as likely in smokers as in nonsmokers. What percentage of women having ectopic pregnancies are smokers?

Accepted Answer

$$\frac{2 	imes 0.32}{2 	imes 0.32 + 0.68} 	imes 100\%$$

Question 12

If $$X is a $$discrete random variable and $$a, b$$ are constants, what is $$E[aX + b]$$?

Accepted Answer

$$aE[X] + b$$

Question 13

Suppose you flip a fair coin $$n$$ times. What is the probability that you get at least one head?

Accepted Answer

$$1 - (1/2)^n$$

Question 14

A random variable $$X$$ has probability mass function $$p(0) = 1/4$$, $$p(1) = 1/2$$, $$p(2) = 1/4. $$What is $$E[X]$$?

Accepted Answer

$$1$$

Question 15

If $$X is a $$binomial random variable with parameters $$n$$ and $$p$$, what is $$Var(X)$$?

Accepted Answer

$$np(1-p)$$

Question 16

A parallel system functions whenever at least one of its $$n$$ components works. Each component works independently with probability $$1/2. $$What is the conditional probability that component $$1$$ works given that the system is functioning?

Accepted Answer

$$\frac{1}{2}$$

Question 17

Suppose you continually collect coupons and there are $$m$$ different types. Each time a new coupon is obtained, it is type $$i$$ with probability $$p_i. $$What is the probability that the $$nth $$coupon is a new type?

Accepted Answer

$$\sum_{i=1}^m p_i \left[1 - P_i\right]$$

Question 18

If $$X is a $$hypergeometric random variable representing the number of white balls selected when $$n$$ balls are randomly chosen from an urn containing $$N$$ balls, $$m of $$which are white, what is $$E[X]$$?

Accepted Answer

$$\frac{nm}{N}$$

Step-by-Step Guidance for Conditional Probability and Random Variables (Statistics)

스터디 가이드 - 연습 문제

학습 가이드 - 플래시카드