Q: Which of the following is the correct formula for the probability of an event $$E$$ using the classical method?

$$P(E) = \frac{\text{number of ways that } E \text{ can occur}}{\text{number of possible outcomes}}$$

Question 1

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

Accepted Answer

As the number of trials in a probability experiment increases, the proportion of times a particular outcome occurs approaches the theoretical probability of that outcome.

Question 2

Suppose you roll a fair six-sided die 1000 times. According to the Law of Large Numbers, what should happen to the relative frequency of rolling a '4' as the number of rolls increases?

Accepted Answer

It should approach $$\frac{1}{6}.$$

Question 3

A random process is defined as:

Accepted Answer

A scenario where the outcome of any particular trial of an experiment is unknown, but the proportion of a particular outcome approaches a specific value as the number of trials increases.

Question 4

In a simulation of 10,000 families where each child has a 0.5 probability of being a girl, what is the expected proportion of families with exactly four girls out of four children?

Accepted Answer

$$0.0625$$

Question 5

Which of the following is the correct formula for the probability of an event $$E$$ using the classical method?

Accepted Answer

$$P(E) = \frac{	ext{number of ways that } E 	ext{ can occur}}{	ext{number of possible outcomes}}$$

Question 6

If the probability of an event $$E is$$ $$0.65$$, how many times would you expect $$E to $$occur in 100 trials?

Accepted Answer

$$65$$

Question 7

Which of the following events would be considered 'unusual' according to the standard cutoff used in probability?

Accepted Answer

An event with probability $$0.03$$

Question 8

A bag contains 12 brown, 10 yellow, 8 red, 20 orange, and 15 green candies. If one candy is randomly selected, what is the probability that it is orange?

Accepted Answer

$$\frac{20}{65}$$

Question 9

In the context of probability, what is a sample space?

Accepted Answer

The collection of all possible outcomes of a probability experiment.

Question 10

Suppose you observe a red light at an intersection on 125 out of 300 days. What is the empirical probability of being stopped by a red light at this intersection?

Accepted Answer

$$\frac{125}{300} = 0.417$$

Question 11

If the probability of an event $$E is$$ $$0$$, what does this mean?

Accepted Answer

The event is impossible.

Question 12

Which rule states that the sum of the probabilities of all outcomes in the sample space must equal 1?

Accepted Answer

Rule 2 of the Rules of Probabilities

Question 13

A probability model lists all possible outcomes and each outcome's probability. Which of the following must be true for a valid probability model?

Accepted Answer

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

Question 14

In a game where a pig is rolled 9,393 times, the outcome 'Side with no dot' occurred 1,244 times. What is the empirical probability of this outcome?

Accepted Answer

$$\frac{1244}{9393} \approx 0.132$$

Question 15

If you flip a fair coin 100 times, what is the expected proportion of heads?

Accepted Answer

$$0.5$$

Question 16

Which of the following statements about the Law of Averages is correct?

Accepted Answer

The Law of Averages is a misconception; past outcomes do not affect future probabilities in random processes.

Question 17

Suppose you have a bag with 12 brown, 10 yellow, 8 red, 20 orange, and 15 green candies. What is the probability of randomly selecting a yellow or green candy?

Accepted Answer

$$\frac{10 + 15}{65} = \frac{25}{65}$$

Question 18

Which of the following best describes an 'event' in probability?

Accepted Answer

Any collection of outcomes from a probability experiment.

Question 19

If the probability of drawing a blue candy from a bag is $$0.23$$, what is the probability of not drawing a blue candy?

Accepted Answer

$$1 - 0.23 = 0.77$$

Question 20

Which of the following is NOT a requirement for using the classical method of probability?

Accepted Answer

Outcomes must be observed empirically.

Probability Rules and Random Processes: Study Notes

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