Question 1

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

Accepted Answer

The mean of the distribution must be an integer.

Question 2

Given the following probability distribution for the number of classes a student takes: $$P(2) = 0.06$$, $$P(3) = 0.11$$, $$P(4) = 0.39$$, $$P(5) = 0.44$$, what is the expected value $$E(x)$$?

Accepted Answer

$$E(x) = 4.21$$

Question 3

A probability distribution is given by $$P(x) = 0.2$$ for $$x=1$$, $$P(x) = 0.5$$ for $$x=2$$, and $$P(x) = 0.3$$ for $$x=3. Is $$this a valid probability distribution?

Accepted Answer

Yes, because all probabilities are between 0 and 1 and sum to 1.

Question 4

Which scenario is best modeled by a binomial distribution?

Accepted Answer

Counting the number of defective items in a fixed sample of 20 products, where each product is either defective or not.

Question 5

Suppose you buy 5 lottery tickets, each with a probability of winning of 0.10. What is the probability that none of your tickets win?

Accepted Answer

$$0.5905$$

Question 6

For a binomial experiment with $$n=5$$ and $$p=0.10$$, what is the expected number of successes?

Accepted Answer

$$0.5$$

Question 7

A concert venue is considering two pricing strategies. In scenario A, the expected profit is $$30 per ticket, and in scenario B, it is 31.50 $$per ticket. Which scenario should the venue choose to maximize long-term profit?

Accepted Answer

Scenario B, because it has a higher expected profit.

Question 8

Which of the following formulas correctly gives the probability of exactly $$k$$ successes in $$n$$ independent Bernoulli trials with probability $$p of $$success?

Accepted Answer

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

Question 9

If $$P(X=0) = 0.5905$$ for a binomial experiment, what is the probability that at least one success occurs?

Accepted Answer

$$0.4095$$

Question 10

A binomial experiment consists of 8 trials with probability of success $$p=0.25. $$What is the probability of exactly 2 successes? (Use the binomial formula.)

Accepted Answer

$$P(X=2) = \frac{8!}{2!6!} (0.25)^2$ (0.75)^6$

Question 11

Which of the following is the correct interpretation of the expected value in a probability distribution?

Accepted Answer

It is the long-run average value of the random variable over many repetitions.

Question 12

Suppose a survey finds that 5% of people will buy premium seats, 20% will buy middle-level seats, and 75% will buy cheap seats at a concert. What is the expected revenue per ticket if the prices are $$100, 50$$, and $20, respectively?

Accepted Answer

$$30$$

Question 13

Which of the following statements about the binomial distribution is FALSE?

Accepted Answer

The number of possible outcomes for each trial must be more than two.

Question 14

If $$P(X=1) = 0.3281$$ and $$P(X=0) = 0.5905$$ for a binomial experiment, what is the probability that at most one success occurs?

Accepted Answer

$$0.9186$$

Question 15

A binomial experiment has $$n=10$$ and $$p=0.3. $$What is the mean (expected value) of the number of successes?

Accepted Answer

$$3$$

Question 16

Which of the following is the correct formula for the expected value $$E(x) of a $$discrete random variable $$x$$ with probability distribution $$P(x)$$?

Accepted Answer

$$E(x) = \sum x \cdot P(x)$$

Question 17

A student claims that in a binomial experiment, the probability of success can change from trial to trial. Is this correct?

Accepted Answer

No, the probability of success must remain constant for all trials.

Question 18

If you want to find the probability of 'at least 2' successes in a binomial experiment, which of the following is the correct complement rule to use?

Accepted Answer

$$P(X \geq 2) = 1 - [P(X=0) + P(X=1)]$$

Question 19

Which of the following is the correct value of $$5! (5 $$factorial)?

Accepted Answer

$$120$$

Discrete Probability Distributions – STAT 201 Study Notes

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