5.1: Probability Rules

Learning Objectives: 1. Apply the rules of probabilities 2. Compute and interpret probabilities using the empirical method 3. Compute and interpret probabilities using the classical method 4. Use simulation to obtain data based on probabilities 5. Recognize and interpret subjective probabilities

1.) Match terms with definitions in the table below:

• Event

• Unusual Event

• Probability

• Sample Space

• Impossible Event

• Equally likely outcomes

• Experiment

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​Word/Phrase	Definition
	The measure of the likelihood of a random phenomenon or chance behavior.
	Any process with uncertain results that can be repeated.
	One or more outcomes from a probability experiment.
	The collection of all possible outcomes.
	Each outcome has the same probability of occurring.
	An event where the probability of occurring is 0.
	An event that has a low probability of occurring

​Word/Phrase	Definition
Probability	The measure of the likelihood of a random phenomenon or chance behavior.
Experiment	Any process with uncertain results that can be repeated.
Event	One or more outcomes from a probability experiment.
Sample space	The collection of all possible outcomes.
Equally likely outcomes	Each outcome has the same probability of occurring.
Impossible event	An event where the probability of occurring is 0.
Unusual event	An event that has a low probability of occurring.

2.) Complete the sentence below:

• “In a probability​ model, the sum of the probabilities of all outcomes must equal…”

1

3.) Answer Parts 1-2.

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1. Is the following a Probability Model?

COLOR	PROBABILITY
Red	0.15
Green	0.2
Blue	0
Brown	0.3
Yellow	0.2
Orange	0.15

2. What do we call the outcome “Blue”?

1. Yes, because the probabilities sum to 1 and they are all greater than or equal to 0, and less than or equal to 1.

2. Impossible Event

4.) Explain why the following is NOT a Probability Model?

COLOR	PROBABILITY
Red	0.3
Green	-0.2
Blue	0.1
Brown	0.4
Yellow	0.2
Orange	0.2

This is not a probability Model because at least 1 probability is less than 1.

5.) Bob is asked to construct a probability model for rolling a pair of fair dice. He lists the outcomes as​ 2, 3,​ 4, 5,​ 6, 7,​ 8, 9,​ 10, 11, 12. Because there are 11​ outcomes, he​ reasoned, the probability of rolling a three must be 1/11. What is wrong with​ Bob's reasoning?

• You always want to assess: Is the situation using Empirical, Subjective, or a Classical Method? Once you’ve figured that out, you may go about solving the question.

• In this example, Bob is attempting to compute the probability using the classical method by dividing the number of ways that the event can occur by the number of possible outcomes. Think about why this method is not appropriate. What does not apply to the Classical method?

The experiment does not have equally likely outcomes.

6.) According to a certain​ country's department of​ education, 39.6% off 3-year-olds are enrolled in daycare. What is the probability that a randomly selected​ 3-year-old is enrolled in day​ care?

• Probability is a measure of the likelihood of a random phenomenon or chance behavior. Probability describes the​ long-term proportion with which a certain outcome will occur in situations with​ short-term uncertainty. To change a percent to a​ decimal, drop the percent sign and move the decimal point two places to the left.

The probability that a randomly selected​ 3-year-old is enrolled in day care is .396

7.) Let the sample space be S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Suppose the outcomes are equally likely. Compute the probability of the event E={3, 10}.

P(E) = 0.2

• all i did was 2/10.

8.) Let the sample space be S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Suppose the outcomes are equally likely. Compute the probability of the event E = “an odd number less than 8”

0.4

• odd numbers < 8 are: 7, 5, 3, and 1. Those are 4 events out of the 10 in the sample space. Thus, 4/10 = 0.4

9.) Let the sample space be S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Suppose the outcomes are equally likely. Compute the probability of the event E = “an odd number”

0.5

• There are 5 odd numbered events. Thus, you divide 5 by the number of events in the sample space (10). 5/10 is… 0.5

10.) A survey of 400 randomly selected high school students determined that 119 play organized sports.

​(a) What is the probability that a randomly selected high school student plays organized​ sports? (round to the nearest thousandth as needed)

​(b) Interpret this probability.

a) 0.298

• Sample size is 400. 119 is the number of people who play sports. To find the probability that a randomly selected individual plays sports, you do 400/119.

  • 119/400 = 0.2975

    • Rounded is 0.298

b) If​ 1,000 high school students were​ sampled, it would be expected that about 298 of them play organized sports.

• So to find the number value of any probability, you just round to the nearest xth place and move the decimal. Or, in this case, the probability was already rounded and the decimal place is in the right spot. Therefore, just take the decimal away, and you get its whole number, which is 298. Most of the time the value is already there, just take away the decimal.

  • Remember, we measure “ABOUT” values in probability. Probabilities give estimates, so when interpreting, make sure you put “ABOUT” before any answer.

11.) In a national survey college students were​ asked, "How often do you wear a seat belt when riding in a car driven by someone​ else?" The response frequencies appear in the table below.​ Answer the questions:

​Response	Frequency
Never	139
Rarely	320
Sometimes	554
Most of the time	1052
Always	2314

a) Construct a probability model for​ seat-belt use by a passenger (Round to the nearest thousandth)

b) Would you consider it unusual to find a college student who never wears a seat belt when riding in a car driven by someone​ else?

a) Rounded to the nearest thousandth, “Never” is about 0.32, “Rarely” is about .073, “Sometimes” is about .127, “Most of the time” is about .240, “Always” is about .528

• When a probability experiment is​ run, probabilities are approximated using the empirical approach. That​ is, the probability of an event E is approximately the number of times event E is observed divided by the number of repetitions of the experiment, i.e., P(E) = the RF of E —> frequency of E/ # of trials of experiment

b) Yes, because P(never) <0.05

• Events closer to 0 are considered unusual.

12.)Determine whether the probabilities are computed using the classical​ method, empirical​ method, or subjective method. Complete parts ​(a) through (d)

** Hint: Classical probability is used when each outcome in a sample space is equally likely. Empirical probability is based on observations obtained from probability experiments. Subjective probability of an outcome is a probability obtained on the basis of personal judgment.

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(a) The probability of having eight even digits on an eight​-digit license plate number is 0.00390625. Choose the correct answer.

A. Subjective method

B. Empirical method

C. Classical method

D. It is impossible to determine which method is used.

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​(b) On the basis of a survey of 1000 families with eight children, the probability of a family having eight girls is 0.00690. Choose the correct answer.

A. Empirical method

B. Subjective method

C. Classical method

D. It is impossible to determine which method is used.

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​(c) According to a sports​ analyst, the probability that a football team will win the next game is 0.45. Choose the correct answer.

A. Classical method

B. Subjective method

C. Empirical method

D. It is impossible to determine which method is used.

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​(d) On the basis of clinical​ trials, the probability of efficacy of a new drug is 0.74. Choose the correct answer.

A. Subjective method

B. Classical method

C. Empirical method

D. It is impossible to determine which method is used.

(a) The probability of having eight even digits on an eight​-digit license plate number is 0.00390625. Choose the correct answer.

A. Subjective method

B. Empirical method

C. Classical method

D. It is impossible to determine which method is used.

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​(b) On the basis of a survey of 1000 families with eight children, the probability of a family having eight girls is 0.00690. Choose the correct answer.

A. Empirical method

B. Subjective method

C. Classical method

D. It is impossible to determine which method is used.

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​(c) According to a sports​ analyst, the probability that a football team will win the next game is 0.45. Choose the correct answer.

A. Classical method

B. Subjective method

C. Empirical method

D. It is impossible to determine which method is used.

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​(d) On the basis of clinical​ trials, the probability of efficacy of a new drug is 0.74. Choose the correct answer.

A. Subjective method

B. Classical method

C. Empirical method

D. It is impossible to determine which method is used.

Probability