Q: A lottery ticket costs $$1 and pays 5000 if $$you win. The probability of winning is $$0.0001. $$What is the expected value of buying one ticket?

$$0.0001 \times 5000 + 0.9999 \times (-1) = -0.50$$

Q: In a binomial experiment with $$n = 5$$ and $$p = 0.2$$, what is the expected number of successes?

$$n \times p = 5 \times 0.2 = 1$$

Question 1

Which of the following best describes a discrete random variable?

Accepted Answer

A variable that assigns a numerical value to each outcome in the sample space and has a countable number of distinct values.

Question 2

A fair coin is tossed three times. What is the probability of getting exactly two heads?

Accepted Answer

$$\frac{3}{8}$$

Question 3

Which formula represents the expected value $$E(X) of a $$discrete random variable $$X$$ with possible values $$x_i$$ and probabilities $$p(x_i)$$?

Accepted Answer

$$E(X) = \sum_{i=1}^{n} x_i \cdot p(x_i)$$

Question 4

A lottery ticket costs $$1 and pays 5000 if $$you win. The probability of winning is $$0.0001. $$What is the expected value of buying one ticket?

Accepted Answer

$$0.0001 	imes 5000 + 0.9999 	imes (-1) = -0.50$$

Question 5

Which of the following is NOT a characteristic of a binomial distribution?

Accepted Answer

Trials are dependent and sampled without replacement.

Question 6

In a binomial experiment with $$n = 5$$ and $$p = 0.2$$, what is the expected number of successes?

Accepted Answer

$$n 	imes p = 5 	imes 0.2 = 1$$

Question 7

Which formula gives the probability of $$x$$ successes in $$n$$ binomial trials with probability $$p of $$success?

Accepted Answer

$$P(X = x) = \frac{n!}{x!(n-x)!} p^x (1-p$$)^{n-x}$$

Question 8

A hypergeometric distribution is used when:

Accepted Answer

Trials are dependent and sampled without replacement from a finite population.

Question 9

Which formula represents the probability of $$x$$ successes in a hypergeometric distribution?

Accepted Answer

$$P(X = x) = \frac{\binom{s}{x} \binom{N-s}{n-x}}{\binom{N}{n}}$$

Question 10

Which scenario best fits the use of a Poisson distribution?

Accepted Answer

Counting the number of defective screws in a box of 5000 screws.

Question 11

Given a Poisson distribution with mean $$\lambda = 2$$, what is the probability of observing exactly 3 events?

Accepted Answer

$$P(X=3) = \frac{e$$^{-2}$$ \cdot 2^3$}{3!}$$

Question 12

Which of the following is a property of the probability mass function (PMF) for a discrete random variable?

Accepted Answer

The sum of all probabilities equals one.

Question 13

A uniform discrete distribution is observed when rolling a fair six-sided die. What is the probability of any single outcome?

Accepted Answer

$$\frac{1}{6}$$

Question 14

Which of the following is NOT a discrete probability distribution?

Accepted Answer

Normal distribution

Question 15

In a binomial experiment with $$n = 4$$ and $$p = 0.1$$, what is the probability of getting exactly 0 successes?

Accepted Answer

$$0.6561$$

Question 16

If the probability of a house fire is $$0.0002$$ and the insurance payout is $$200,000$$, what is the expected payout per policy?

Accepted Answer

$$0.0002 	imes 200,000 = 40$$

Question 17

Which formula represents the variance $$\$$sigma^2$$ of a binomial distribution with parameters n$ and p$?

Accepted Answer

$$\$$sigma^2$$ = n p (1-p)$$

Question 18

A sample of 5 vouchers is drawn from a population of 200, of which 20 contain errors. Which distribution should be used to find the probability that the sample contains at least two erroneous vouchers?

Accepted Answer

Hypergeometric distribution

Question 19

Which of the following statements about the Poisson distribution is TRUE?

Accepted Answer

It is used to model the number of rare events occurring in a fixed interval of time or space.

Discrete Probability Distributions: Study Notes for Statistics for Business

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