Q: If two events are independent, which of the following is true?

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

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

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

Question 1

A simulation of rolling a 10-sided die 40 times resulted in a '4' being rolled 6 times. What is the empirical probability of rolling a '4' based on this simulation? Express your answer as a decimal.

Accepted Answer

$$0.15$$

Question 2

According to the Law of Large Numbers, what happens to the relative frequency of an event as the number of trials increases?

Accepted Answer

It approaches the theoretical probability.

Question 3

In a simulation where 10,000 families each have four children, the probability that the fifth child is a boy, given the first four are girls, is approximately:

Accepted Answer

$$0.50$$

Question 4

Which of the following best describes the Law of Large Numbers?

Accepted Answer

The proportion of times an event occurs in repeated trials approaches the actual probability as the number of trials increases.

Question 5

Given the Venn diagram where $$E = \{1,3,5,7,9\}$$ and $$F = \{0,1,2,3,4\}$$, what is $$P(E 	ext{ or } F) if $$the sample space $$S = \{0,1,2,3,4,5,6,7,8,9\}$$?

Accepted Answer

$$\frac{8}{10} = 0.8$$

Question 6

If $$P(E) = 0.32$$, $$P(F) = 0.61$$, and $$P(E 	ext{ and } F) = 0.10$$, what is $$P(E 	ext{ or } F)$$?

Accepted Answer

$$0.32 + 0.61 - 0.10 = 0.83$$

Question 7

The complement of event $$E is $$denoted $$E^c. If$$ $$P(E) = 0.316$$, what is $$P(E^c)$$?

Accepted Answer

$$1 - 0.316 = 0.684$$

Question 8

A travel survey shows 4,895 people have a commute time of 90 or more minutes out of a total of 320,000 surveyed. What is the probability a randomly selected person has a commute time of 90 or more minutes?

Accepted Answer

$$\frac{4,895}{320,000} \approx 0.0153$$

Question 9

Which of the following is an example of a subjective probability?

Accepted Answer

An economist predicting a 20% chance of recession next year.

Question 10

If two events are independent, which of the following is true?

Accepted Answer

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

Question 11

A restaurant offers 2 appetizers, 4 entrées, and 2 desserts. Using the multiplication rule of counting, how many different meals can be ordered?

Accepted Answer

$$2 	imes 4 	imes 2 = 16$$

Question 12

How many ways can the letters in the word 'MISSISSIPPI' be arranged?

Accepted Answer

$$\frac{11!}{4!4!2!1!}$$

Question 13

A pizza shop offers 8 toppings. What is the probability that a randomly selected pizza with 2 toppings includes pepperoni, assuming all combinations are equally likely?

Accepted Answer

$$\frac{7}{28} = 0.25$$

Question 14

Which counting technique should you use to determine the number of ways to select 3 marbles from a jar containing 4 red, 5 blue, and 3 yellow marbles, if order does not matter?

Accepted Answer

Combination

Question 15

If $$P(E|F) = 0.4$$ and $$P(F) = 0.5$$, what is $$P(E 	ext{ and } F)$$?

Accepted Answer

$$0.4 	imes 0.5 = 0.2$$

Question 16

A probability model must satisfy which two rules?

Accepted Answer

All probabilities must be between 0 and 1; the sum of all probabilities must equal 1.

Question 17

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

Accepted Answer

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

Question 18

If you are asked to find the probability of a single event and the outcomes are equally likely, which formula should you use?

Accepted Answer

$$P(E) = \frac{N(E)}{N(S)}$$

Chapter 5: Probability – Rules, Methods, and Applications

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