Question 1

A company is analyzing the probability of two independent events: winning a contract in London (probability 0.4) and winning a contract in New York (probability 0.25). What is the probability that the company will win both contracts?

Accepted Answer

$$0.10$$

Question 2

Suppose you flip a fair coin three times. What is the probability of getting exactly two heads?

Accepted Answer

$$0.375$$

Question 3

A deck contains 52 cards. What is the probability of drawing a red card or an ace in a single draw?

Accepted Answer

$$\frac{28}{52}$$

Question 4

A Starbucks manager observes that out of 2,250 drink sales, 1,570 were for caffeinated drinks. Using the relative frequency assessment method, what is the estimated probability that the next customer will order a caffeinated drink?

Accepted Answer

$$0.698$$

Question 5

If two events $$A$$ and $$B$$ are mutually exclusive, which of the following is true?

Accepted Answer

$$P(A \cap B) = 0$$

Question 6

A basketball team has 10 players: 5 seniors, 2 juniors, and 3 sophomores. Two players are randomly selected to be captains. What is the probability that both are seniors?

Accepted Answer

$$0.22$$

Question 7

Which of the following is the correct formula for the expected value (mean) of a discrete random variable $$X$$ with possible values $$x_i$$ and probabilities $$P(x_i)$$?

Accepted Answer

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

Question 8

A binomial experiment consists of $$n=8$$ trials with probability of success $$p=0.37. $$What is the expected number of successes?

Accepted Answer

$$2.96$$

Question 9

Which of the following is NOT an assumption of the binomial distribution?

Accepted Answer

The probability of success changes from trial to trial.

Question 10

A bank receives on average 16 customers per hour. What is the probability that exactly 12 customers arrive in one hour, according to the Poisson distribution?

Accepted Answer

$$P(X=12) = \frac{16$$^{12}$$ e$$^{-16}$$}{12!}$$

Question 11

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

Accepted Answer

The probability of any exact value is greater than zero.

Question 12

Given a normal distribution with mean $$\mu = 100$$ and standard deviation $$\sigma = 20$$, what is the $$z-$$score for $$X = 130$$?

Accepted Answer

$$z = 1.5$$

Question 13

A continuous random variable $$X is $$uniformly distributed between $$a=4$$ and $$b=10. $$What is the probability that $$X is $$between $$4.5$$ and $$9$$?

Accepted Answer

$$0.75$$

Question 14

Which of the following is the correct formula for the variance of a binomial distribution with $$n$$ trials and probability of success $$p$$?

Accepted Answer

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

Question 15

A random variable $$X$$ follows an exponential distribution with mean time between events of $$5$$ minutes. What is the value of $$\lambda$$?

Accepted Answer

$$\lambda = 0.2$$

Question 16

A company has 30 employees (12 women, 18 men) and randomly lays off 10. What is the probability that exactly 8 of those laid off are women? (Use the hypergeometric formula.)

Accepted Answer

$$P = \frac{\binom{12}{8} \binom{18}{2}}{\binom{30}{10}}$$

Question 17

Which of the following scenarios is best modeled by the Poisson distribution?

Accepted Answer

The number of emails received per hour at a help desk.

Question 18

If $$X is a $$continuous random variable with a normal distribution, which of the following is true?

Accepted Answer

$$P(X \u003c a) = P(X \leq a)$$ for any $$a$$

Question 19

A random variable $$X is $$exponentially distributed with $$\lambda = 0.5. $$What is the probability that $$X is $$less than 2?

Accepted Answer

$$1 - e$$^{-0.5 	imes 2}$$

Question 20

Which of the following best describes when to use the hypergeometric distribution?

Accepted Answer

When sampling is without replacement and the probability of success changes from trial to trial.

Probability and Probability Distributions: Study Notes for Business Statistics

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