Q: Given the Venn diagram, if $$E = \{1,3,5,7,9\}$$ and $$F = \{0,1,2,3,4\}$$, what is $$P(E \text{ or } F)$$ for a random selection from $$S = \{0,1,2,3,4,5,6,7,8,9\}$$?

$$P(E \text{ or } F) = \frac{5}{10} + \frac{5}{10} - \frac{2}{10} = 0.8$$

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

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

Q: Which probability rule should you use to find the probability that at least one of 500 randomly selected 60-year-old females will die during the year, given the probability of surviving is 0.992?

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

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?

Accepted Answer

$$P(4) = \frac{6}{40} = 0.15$$

Question 2

According to the Law of Large Numbers, what happens to the relative frequency of rolling a '4' as the number of rolls increases in a fair 10-sided die simulation?

Accepted Answer

It approaches the theoretical probability of $$\frac{1}{10}$$

Question 3

In the table showing the proportion of days stopped at a red light after 15 days, the aggregate proportion is approximately 0.47. What does this value represent?

Accepted Answer

The empirical probability of being stopped at a red light after 15 days

Question 4

A family has four girls in a row. According to the simulation, what is the probability that the fifth child is a boy?

Accepted Answer

$$0.5066$$

Question 5

Which of the following best describes the Law of Averages as discussed in the materials?

Accepted Answer

A common misconception that past outcomes affect future probabilities in random processes

Question 6

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

Accepted Answer

$$P(E 	ext{ or } F) = \frac{5}{10} + \frac{5}{10} - \frac{2}{10} = 0.8$$

Question 7

Which formula should you use to find the probability that a randomly selected resident of Hartford County, CT has a commute time less than 90 minutes, given the table of travel times?

Accepted Answer

$$P(E) = 1 - P(E^c)$$, where $$E^c is $$the event '90 or more minutes'

Question 8

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

Accepted Answer

$$P(E 	ext{ and } F) = P(E) \cdot P(F)$$

Question 9

A pizza shop offers 8 toppings. How many different two-topping pizzas can be made if the order of toppings does not matter?

Accepted Answer

$$C(8,2) = \frac{8!}{2!6!} = 28$$

Question 10

Which probability rule should you use to find the probability that at least one of 500 randomly selected 60-year-old females will die during the year, given the probability of surviving is 0.992?

Accepted Answer

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

Question 11

A probability model must satisfy which two conditions?

Accepted Answer

All probabilities are between 0 and 1, and the sum of all probabilities is 1

Question 12

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

Accepted Answer

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

Question 13

A professor selects 10 questions from a bank of 30 and wants to know in how many ways he can arrange them on an exam. Which formula should he use?

Accepted Answer

$$P(30,10) = \frac{30!}{(30-10)!}$$

Question 14

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

Accepted Answer

An economist predicts a 20% chance of recession next year based on personal judgment

Question 15

If you want to find the probability that a randomly selected household has high-speed internet given that it has land-line phone service, which formula should you use?

Accepted Answer

$$P(	ext{High-speed internet} | 	ext{Land-line}) = \frac{P(	ext{High-speed internet and Land-line})}{P(	ext{Land-line})}$$

Question 16

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

Accepted Answer

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

Question 17

Which of the following best describes the complement of an event $$E$$?

Accepted Answer

All outcomes in the sample space that are not in $$E$$

Chapter 5: Probability – Rules, Methods, and Applications

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