Q: How many different license plates can be made consisting of 3 letters followed by 3 digits, if repetition is allowed?

$$26^3$$ \times $$10^3 = 17$$,576,000$$

Q: You are dealt two cards successively without replacement from a standard deck of 52 playing cards. What is the probability that the first card is a two and the second card is a ten?

$$\frac{4}{52} \times \frac{4}{51} \approx 0.006$$

Q: A binomial distribution has $$n = 600$$ and $$p = 0.87. $$What is the variance of the distribution?

$$\$$sigma^2$$ = n p (1-p) = 600 \times 0.87 \times 0.13 = 67.86$$

Question 1

A standard deck of 52 playing cards is shuffled. What is the probability of drawing a heart from the deck?

Accepted Answer

$$\frac{1}{4}$$

Question 2

A sports team has a three-game road trip. What is the probability that they win exactly two games, assuming each game is independent and the probability of winning any game is $$0.5$$?

Accepted Answer

$$0.375$$

Question 3

How many different license plates can be made consisting of 3 letters followed by 3 digits, if repetition is allowed?

Accepted Answer

$$26^3$$ 	imes $$10^3 = 17$$,576,000$$

Question 4

A group of 100 students is surveyed about credit card ownership. If 22 freshmen own a credit card and there are 60 freshmen in total, what is the probability that a randomly selected freshman owns a credit card?

Accepted Answer

$$\frac{22}{60} \approx 0.367$$

Question 5

You are dealt two cards successively without replacement from a standard deck of 52 playing cards. What is the probability that the first card is a two and the second card is a ten?

Accepted Answer

$$\frac{4}{52} 	imes \frac{4}{51} \approx 0.006$$

Question 6

A test consists of 30 multiple choice questions, each with five possible answers, only one of which is correct. What is the mean and standard deviation of the number of correct answers if random guesses are made?

Accepted Answer

mean: $$6$$, standard deviation: $$2.19$$

Question 7

Which of the following cannot be a probability?

Accepted Answer

$$\frac{4}{3}$$

Question 8

A binomial experiment has $$n = 10$$ trials and probability of success $$p = 0.2. $$What is the mean of the binomial distribution?

Accepted Answer

$$\mu = n 	imes p = 2$$

Question 9

A tourist in Ireland wants to visit eight different cities. If the route is randomly selected, what is the probability that the tourist will visit the cities in alphabetical order?

Accepted Answer

$$\frac{1}{8!} \approx 0.00002$$

Question 10

The random variable $$x$$ represents the number of boys in a family of three children. Assuming boys and girls are equally likely, and $$P(x) is $$given as $$0.125$$ for $$x=0$$ and $$x=3$$, $$0.375$$ for $$x=1$$ and $$x=2. $$What is the mean and standard deviation for $$x$$?

Accepted Answer

mean: $$1.50$$, standard deviation: $$0.87$$

Question 11

A group of college students is surveyed about smoking habits. If 87 are regular smokers and 11 are heavy smokers out of 420 students, what is the probability of randomly selecting a student who is a regular or heavy smoker?

Accepted Answer

$$\frac{87+11}{420} = \frac{98}{420} \approx 0.233$$

Question 12

A binomial distribution has $$n = 600$$ and $$p = 0.87. $$What is the variance of the distribution?

Accepted Answer

$$\$$sigma^2$$ = n p (1-p) = 600 	imes 0.87 	imes 0.13 = 67.86$$

Question 13

A card is drawn from a standard deck of 52 playing cards. What is the probability that the card is an ace or a king?

Accepted Answer

$$\frac{8}{52} = \frac{2}{13}$$

Question 14

A certain raffle sells 10,000 tickets at $$10 each for three prizes valued at 4$$,800$$, 1$$,200$$, and 400$. What is the expected value of one ticket?

Accepted Answer

$$E = \frac{4800+1200+400}{10000} - 10 = -9.36$$

Question 15

Which of the following statements best describes a classical probability?

Accepted Answer

The probability that a newborn baby is a boy is $$\frac{1}{2}.$$

Question 16

A die is rolled. Event A: The result is a 3. Event B: The result is an odd number. Are these events mutually exclusive?

Accepted Answer

Not mutually exclusive

Question 17

A binomial experiment has $$n = 500$$ and $$p = 0.89. $$What is the standard deviation?

Accepted Answer

$$\sigma = \sqrt{n p (1-p)} = \sqrt{500 	imes 0.89 	imes 0.11} \approx 7$$

Question 18

A code consists of 4 digits. The first digit must be 3, 4, or 5, and the code may not end in 0. How many different codes are possible if repetition is allowed?

Accepted Answer

$$3 	imes 10 	imes 10 	imes 9 = 2700$$

Probability, Counting, and Binomial Distribution Study Guide

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