Chapter 4

Probability

Definitions

• A probability experiment is a chance process that leads to well-defined results called outcomes.

• An outcome is the result of a single trial of a probability experiment.

• A sample space is the set of all possible outcomes of a probability experiment.

The probability of any event E is

€

P(E )=#of waysE canhappen

#of elementsinsamplespace

• The probability of an event is a number (either a fraction or a decimal) between zero and one, inclusive.

€

0≤P(E) ≤1

Cardinal Rule

• Never Count Anything Twice!!!

• If the probability of an event is zero, the event is impossible.

• If the probability of an event is one, the event is certain.

• The sum of the probabilities of the outcomes in the sample space is one.

Cardinal Rule

• Never Count Anything Twice!!!

Example 1

• Roll a fair die.– Find the sample space.

Example 2

• A family wants to have three children.– How many elements are in the sample space?– List the elements in the sample space.

Example 1

• Roll a fair die.– Find the sample space.– Find the probability of rolling an even number.– Find the probability of rolling a number that is

prime.

Example 2

• A family wants to have three children.– How many elements are in the sample space?– List the elements in the sample space.– Find the probability that the family has one girl.– Find the probability that the family has at most

one girl.– Find the probability that the family has at least

one boy.

Example 3

• A card is drawn from an ordinary deck of cards.– Find the probability of getting a jack.– Find the probability of getting the 6 of clubs.– Find the probability of getting a red card.– Find the probability of getting a red jack.– Find the probability of getting a red card or a

jack.– Find the probability of getting a jack and a

king.

Cardinal Rule

• Never Count Anything Twice!!!

• Two events are mutually exclusive if they cannot occur at the same time.

Example 4

• A jar contains 3 red, 2 blue, 4 green, and 1 white marble. A marble is chosen randomly.– Find the probability of choosing a red marble.– Find the probability of choosing a red or a blue

marble.– Find the probability of choosing a red and a

blue marble.

Example 5• A jar contains 3 red, 2 blue, 4 green, and 1

white marble. Two marbles are randomly chosen, choosing the 2nd marble after replacing the first.– Find the probability of choosing 2 red marbles.– Find the probability of choosing a red and then

a green marble.– Find the probability of choosing a red and a

green marble.– Find the probability of choosing two white

marbles.

• Two events A and B are independent if the fact that A occurs does not affect the probability of B occurring.

Example 6

• A jar contains 3 red, 2 blue, 4 green, and 1 white marble. Two marbles are randomly chosen, choosing the 2nd marble without replacing the first.– Find the probability of choosing 2 red marbles.– Find the probability of choosing a red marble

and then a green marble.– Find the probability of choosing two white

marbles.

• When the outcome or occurrence of the first events affects the outcome or occurrence of the second event in such a way that the probability is changes, the events are said to be dependent events.

Example 7

• If 25% of U.S. federal prison inmates are not U.S. citizens, find the probability that two randomly selected federal prison inmates will not be U. S. citizens.

Example 8

• A flashlight has 6 batteries, two of which are defective. If two batteries are randomly selected, find the probability that both are defective.

Example 9

• In a large shopping mall, a marketing agency conducted a survey on credit cards. The results are below.

Employment Status

Owns a credit card

Does not own a credit card

Employed 18 29

Unemployed 28 34

Cardinal Rule

• Never Count Anything Twice!!!

• If one person is randomly selected,– Find the probability that the person owns a

credit card.– Find the probability that the person is

employed.– Find the probability that the person owns a

credit card and is employed.– Find the probability that the person owns a

credit card or is employed.– Find the probability that the person owns a

credit card given that the person is employed.– Find the probability that the person is employed

given that the person owns a credit card.

• If you see the word “given” in a problem, the sample space has changed.

• The new sample space is what comes after the word “given”

• In other words, in our formula for probability, the denominator changes to what comes after the word “given”

Cardinal Rule

• Never Count Anything Twice!!!

• Are the events “owning a credit card” and “being employed” mutually exclusive events? Why or why not?