Question 1

Which of the following best describes a discrete random variable?

Accepted Answer

A variable whose values are determined by chance and can only take on a finite or countable number of values.

Question 2

Given the probability distribution for the number of heads when tossing a coin twice: $$X = \{0, 1, 2\}$$ with $$P(X=0) = 0.25$$, $$P(X=1) = 0.5$$, $$P(X=2) = 0.25$$, what is the expected value (mean) of $$X$$?

Accepted Answer

$$E[X] = 1$$

Question 3

Which of the following is NOT a requirement for a valid probability distribution for a discrete random variable?

Accepted Answer

The random variable must be continuous.

Question 4

Suppose $$X is $$the number of heads when tossing a coin three times. What is $$P(X \geq 2)$$?

Accepted Answer

$$0.5$$

Question 5

A probability distribution has the following probabilities: $$P(X=0) = 0.3$$, $$P(X=1) = 0.4$$, $$P(X=2) = 0.2$$, $$P(X=3) = 0.1. Is $$this a valid probability distribution?

Accepted Answer

Yes, because the sum of the probabilities is 1.

Question 6

Which of the following scenarios describes a binomial experiment?

Accepted Answer

Counting the number of defective items in a sample of 20 from a production line.

Question 7

If $$X \sim 	ext{Bin}(n=8, p=0.45)$$, what is the mean (expected value) of $$X$$?

Accepted Answer

$$\mu = 3.6$$

Question 8

A game has the following probability distribution for winnings $$X$$: $$P(X=-5)=0.2$$, $$P(X=0)=0.2$$, $$P(X=5)=0.2$$, $$P(X=10)=0.2$$, $$P(X=15)=0.2. $$What is the expected value of your winnings?

Accepted Answer

$$E[X] = 5$$

Question 9

For a binomial distribution with $$n=10$$ and $$p=0.5$$, what is the standard deviation?

Accepted Answer

$$\sigma = 1.58$$

Question 10

Suppose the probability that a randomly selected adult in the US is infected with HIV is $$0.01. In a $$sample of $$n=30$$ adults, what is the variance of the number of infected people?

Accepted Answer

$$\$$sigma^2$$ = 0.3$$

Question 11

Which of the following is a property of a probability mass function (p.m.f.) for a discrete random variable $$X$$?

Accepted Answer

$$0 \leq P(X=x) \leq 1$$ for all $$x$$

Question 12

A probability histogram is different from a relative frequency histogram in that:

Accepted Answer

The vertical axis shows probabilities instead of relative frequencies.

Question 13

If $$X is a $$binomial random variable with $$n=20$$ and $$p=0.3$$, what is the probability of exactly 5 successes? (Use the binomial formula)

Accepted Answer

$$P(X=5) = \dfrac{20!}{15!5!} (0.3)^5$ (0.7$$)^{15}$$

Question 14

A probability distribution has $$\mu = 5$$ and $$\sigma = 1.58. $$According to the range rule of thumb, what is the maximum value that is not considered significantly high?

Accepted Answer

$$8.16$$

Question 15

Which of the following situations would NOT result in a binomial distribution?

Accepted Answer

Recording the number of televisions in 50 households.

Question 16

If $$X is a $$discrete random variable with the following probability distribution: $$P(X=1)=0.1$$, $$P(X=2)=0.3$$, $$P(X=3)=0.4$$, $$P(X=4)=0.2$$, what is $$P(X\u003e1)$$?

Accepted Answer

$$0.7$$

Question 17

A family plans to have 10 children. If the probability of having a girl is $$0.5$$, what is the probability that exactly 5 of the children are girls? (Use the binomial formula)

Accepted Answer

$$P(X=5) = \dfrac{10!}{5!5!} (0.5)^5$ (0.5)^5$

Question 18

If the probability of a randomly selected blood donor having type O-negative blood is $$0.45$$, and 8 donors are selected, what is the mean number of donors with type O-negative blood?

Accepted Answer

$$3.6$$

Question 19

In a binomial distribution, which formula gives the standard deviation?

Accepted Answer

$$\sigma = \sqrt{np(1-p)}$$

5 Discrete Probability Distributions and the Binomial Distribution

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