Example

Klaus is trying to choose where to go on vacation. His two choices are: [latex]A[/latex] = New Zealand and [latex]B[/latex] = Alaska.

• Klaus can only afford one vacation. The probability that he chooses [latex]A[/latex] is [latex]P(A) = 0.6[/latex] and the probability that he chooses [latex]B[/latex] is [latex]P(B) = 0.35[/latex].
• [latex]P(A \text{ AND } B) = 0[/latex] because Klaus can only afford to take one vacation.
• Therefore, the probability that he chooses either New Zealand or Alaska is [latex]P(A \text{ OR } B) = P(A) + P(B) = 0.6 + 0.35 = 0.95[/latex]. Note that the probability that he does not choose to go anywhere on vacation must be [latex]0.05[/latex].

Example

Carlos plays college soccer. He makes a goal [latex]65[/latex]% of the time he shoots. Carlos is going to attempt two goals in a row in the next game. [latex]A =[/latex] the event Carlos is successful on his first attempt. [latex]P(A) = 0.65. B = [/latex] the event Carlos is successful on his second attempt. [latex]P(B) = 0.65[/latex]. Carlos tends to shoot in streaks. The probability that he makes the second goal GIVEN that he made the first goal is [latex]0.90[/latex].

1. What is the probability that he makes both goals?
2. What is the probability that Carlos makes either the first goal or the second goal?
3. Are [latex]A[/latex] and [latex]B[/latex] independent?
4. Are [latex]A[/latex] and [latex]B[/latex] mutually exclusive?

Example

A community swim team has 150 members. Seventy-five of the members are advanced swimmers. Forty-seven of the members are intermediate swimmers. The remainder are novice swimmers. Forty of the advanced swimmers practice four times a week. Thirty of the intermediate swimmers practice four times a week. Ten of the novice swimmers practice four times a week. Suppose one member of the swim team is chosen randomly.

1. What is the probability that the member is a novice swimmer?
2. What is the probability that the member practices four times a week?
3. What is the probability that the member is an advanced swimmer and practices four times a week?
4. What is the probability that a member is an advanced swimmer and an intermediate swimmer? Are being an advanced swimmer and an intermediate swimmer mutually exclusive? Why or why not?
5. Are being a novice swimmer and practicing four times a week independent events? Why or why not?

Example

Felicity attends Modesto JC in Modesto, CA. The probability that Felicity enrolls in a math class is 0.2 and the probability that she enrolls in a speech class is 0.65. The probability that she enrolls in a math class, given that she enrolls in speech class, is 0.25.

Let [latex]M[/latex] = math class, [latex]S[/latex] = speech class, [latex]M|S[/latex] = math given speech.

1. What is the probability that Felicity enrolls in math and speech? Find [latex]P(M \text{ AND } S) = P(M|S)P(S)[/latex].
2. What is the probability that Felicity enrolls in math or speech classes? Find [latex]P(M \text{ OR } S) = P(M) + P(S) - P(M \text{ AND } S)[/latex].
3. Are [latex]M[/latex] and [latex]S[/latex] independent? Is [latex]P(M|S) = P(M)[/latex]?
4. Are [latex]M[/latex] and [latex]S[/latex] mutually exclusive? Is [latex]P(M \text{ AND } S) = 0[/latex]?

Example

Studies show that about one woman in seven (approximately [latex]14.3[/latex]%) who live to be [latex]90[/latex] will develop breast cancer. Suppose that of those women who develop breast cancer, a test is negative [latex]2[/latex]% of the time. Also suppose that in the general population of women, the test for breast cancer is negative about [latex]85[/latex]% of the time. Let [latex]B[/latex] = woman develops breast cancer and let [latex]N[/latex] = tests negative. Suppose one woman is selected at random.

1. What is the probability that the woman develops breast cancer? What is the probability that woman tests negative?
2. Given that the woman has breast cancer, what is the probability that she tests negative?
3. What is the probability that the woman has breast cancer AND tests negative?
4. What is the probability that the woman has breast cancer or tests negative?
5. Are having breast cancer and testing negative independent events?
6. Are having breast cancer and testing negative mutually exclusive?

Example

Refer to the information in the earlier example on women who develop breast cancer. [latex]P[/latex] = tests positive.

1. Given that a woman develops breast cancer, what is the probability that she tests positive. Find [latex]P(P|B) = 1 – P(N|B)[/latex].
2. What is the probability that a woman develops breast cancer and tests positive? Find [latex]P(B \text{ AND } P) = P(P|B)P(B)[/latex].
3. What is the probability that a woman does not develop breast cancer? Find [latex]P(B') = 1 – P(B)[/latex].
4. What is the probability that a woman tests positive for breast cancer? Find [latex]P(P) = 1 – P(N)[/latex].