Q: Given the travel survey data (Drive alone: 83, Carpool: 22, Public transportation: 10, Walk: 5, Other means: 3, Work at home: 77), what is the empirical probability that a randomly selected individual carpools to work?

$$P(\text{Carpool}) = \frac{22}{200} = 0.11$$

Q: If you draw a single card from a standard 52-card deck, what is the probability of drawing a king or a queen or a jack?

$$P(\text{King or Queen or Jack}) = \frac{12}{52}$$

Q: Using the general addition rule, what is $$P(E \text{ or } F) if$$ $$P(E) = 0.3$$, $$P(F) = 0.4$$, and $$P(E \text{ and } F) = 0.1$$?

$$P(E \text{ or } F) = 0.3 + 0.4 - 0.1 = 0.6$$

Q: From the marital status table (total population 271.7 million), what is the probability that a randomly selected U.S. resident 15 years or older is widowed or divorced?

$$P(\text{Widowed or Divorced}) = \frac{15.0 + 28.9}{271.7}$$

Q: Which formula should you use to find the probability of at least one event occurring in $$n$$ independent trials?

$$P(\text{at least one}) = 1 - P(\text{none})$$

Q: If $$P(E) = 0.4$$ and $$P(F|E) = 0.5$$, what is $$P(E \text{ and } F)$$ using the general multiplication rule?

$$P(E \text{ and } F) = 0.4 \times 0.5 = 0.2$$

Question 1

A coin is flipped 100 times. The proportion of heads after each flip is plotted. What does the long-term behavior of the proportion of heads illustrate about probability?

Accepted Answer

As the number of flips increases, the proportion of heads approaches the theoretical probability of 0.5.

Question 2

Given the travel survey data (Drive alone: 83, Carpool: 22, Public transportation: 10, Walk: 5, Other means: 3, Work at home: 77), what is the empirical probability that a randomly selected individual carpools to work?

Accepted Answer

$$P(	ext{Carpool}) = \frac{22}{200} = 0.11$$

Question 3

Which of the following is a valid probability model for a single six-sided die?

Accepted Answer

Each side has probability $$\frac{1}{6}$$ and the sum of all probabilities is 1.

Question 4

If you draw a single card from a standard 52-card deck, what is the probability of drawing a king or a queen or a jack?

Accepted Answer

$$P(	ext{King or Queen or Jack}) = \frac{12}{52}$$

Question 5

Suppose the probability that a randomly selected 24-year-old male will survive the year is 0.9986. What is the probability that three randomly selected 24-year-old males will all survive the year?

Accepted Answer

$$0.9986^3$$

Question 6

Using the general addition rule, what is $$P(E 	ext{ or } F) if$$ $$P(E) = 0.3$$, $$P(F) = 0.4$$, and $$P(E 	ext{ and } F) = 0.1$$?

Accepted Answer

$$P(E 	ext{ or } F) = 0.3 + 0.4 - 0.1 = 0.6$$

Question 7

If the probability that a bulb will last 2 years is 0.995 and the bulbs are independent, what is the probability that at least one bulb will burn out in a string of 100 bulbs?

Accepted Answer

$$1 - 0.995$$^{100}$$

Question 8

From the marital status table (total population 271.7 million), what is the probability that a randomly selected U.S. resident 15 years or older is widowed or divorced?

Accepted Answer

$$P(	ext{Widowed or Divorced}) = \frac{15.0 + 28.9}{271.7}$$

Question 9

Which formula should you use to find the probability of at least one event occurring in $$n$$ independent trials?

Accepted Answer

$$P(	ext{at least one}) = 1 - P(	ext{none})$$

Question 10

A committee of 3 is to be formed from 14 people, with the positions of chair, vice-chair, and secretary. How many different ways can the committee be formed?

Accepted Answer

$$14 	imes 13 	imes 12$$

Question 11

If you randomly select 2 friends from a group of 4 to attend a concert, and order does not matter, how many possible pairs can be selected?

Accepted Answer

$$\binom{4}{2} = 6$$

Question 12

Which of the following best describes the classical method of probability?

Accepted Answer

It is based on counting equally likely outcomes without performing the experiment.

Question 13

Suppose $$P(E) = 0.7$$ and $$P(E^c) is $$the probability of the complement of $$E. $$What is $$P(E^c)$$?

Accepted Answer

$$P(E^c) = 1 - 0.7 = 0.3$$

Question 14

If $$P(E) = 0.4$$ and $$P(F|E) = 0.5$$, what is $$P(E 	ext{ and } F)$$ using the general multiplication rule?

Accepted Answer

$$P(E 	ext{ and } F) = 0.4 	imes 0.5 = 0.2$$

Question 15

Which of the following scenarios best illustrates the use of subjective probability?

Accepted Answer

An economist estimates the chance of a recession next year.

Question 16

If you are dealt 4 cards from a standard deck of 52, what is the probability that all 4 cards are aces?

Accepted Answer

$$\frac{\binom{4}{4}}{\binom{52}{4}}$$

Question 17

Which formula would you use to calculate the number of ways to arrange 5 books on a shelf from a collection of 7 books?

Accepted Answer

$$P(7,5) = \frac{7!}{(7-5)!}$$

Probability Rules and Counting Techniques – Study Notes for Statistics

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