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 $\lambda > 0$. 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 $n$th 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)

Study Guide - Practice Questions

Study Guide - Flashcards