# 3.3 Two basic rules of probability  (Page 2/26)

 Page 2 / 26

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.

a. What is the probability that the member is a novice swimmer?

a. $\frac{28}{150}$

b. What is the probability that the member practices four times a week?

b. $\frac{80}{150}$

c. What is the probability that the member is an advanced swimmer and practices four times a week?

c. $\frac{40}{150}$

d. 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?

d. P (advanced $\cap$ intermediate) = 0, so these are mutually exclusive events. A swimmer cannot be an advanced swimmer and an intermediate swimmer at the same time.

e. Are being a novice swimmer and practicing four times a week independent events? Why or why not?

e. No, these are not independent events.
P (novice $\cap$ practices four times per week) = 0.0667
P (novice) P (practices four times per week) = 0.0996
0.0667 ≠ 0.0996

## Try it

A school has 200 seniors of whom 140 will be going to college next year. Forty will be going directly to work. The remainder are taking a gap year. Fifty of the seniors going to college play sports. Thirty of the seniors going directly to work play sports. Five of the seniors taking a gap year play sports. What is the probability that a senior is taking a gap year?

$P=\frac{200-140-40}{200}=\frac{20}{200}=0.1$

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 $|$ that she enrolls in speech class is 0.25.

Let: M = math class, S = speech class, M $|$ S = math given speech

1. What is the probability that Felicity enrolls in math and speech?
Find P ( M $\cap$ S ) = P ( M $|$ S ) P ( S ).
2. What is the probability that Felicity enrolls in math or speech classes?
Find P ( M $\cup$ S ) = P ( M ) + P ( S ) - P ( M $\cap$ S ).
3. Are M and S independent? Is P ( M $|$ S ) = P ( M )?
4. Are M and S mutually exclusive? Is P ( M $\cap$ S ) = 0?

a. 0.1625, b. 0.6875, c. No, d. No

## Try it

A student goes to the library. Let events B = the student checks out a book and D = the student check out a DVD. Suppose that P ( B ) = 0.40, P ( D ) = 0.30 and P ( D $|$ B ) = 0.5.

1. Find P ( B $\cap$ D ).
2. Find P ( B $\cup$ D ).
1. P ( B $\cap$ D ) = P ( D $|$ B ) P ( B ) = (0.5)(0.4) = 0.20.
2. P ( B $\cup$ D ) = P ( B ) + P ( D ) − P ( B $\cap$ D ) = 0.40 + 0.30 − 0.20 = 0.50

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

a. What is the probability that the woman develops breast cancer? What is the probability that woman tests negative?

a. P ( B ) = 0.143; P ( N ) = 0.85

b. Given that the woman has breast cancer, what is the probability that she tests negative?

b. P ( N $|$ B ) = 0.02

c. What is the probability that the woman has breast cancer AND tests negative?

c. P ( B $\cap$ N ) = P ( B ) P ( N $|$ B ) = (0.143)(0.02) = 0.0029

d. What is the probability that the woman has breast cancer or tests negative?

d. P ( B $\cup$ N ) = P ( B ) + P ( N ) - P ( B $\cap$ N ) = 0.143 + 0.85 - 0.0029 = 0.9901

e. Are having breast cancer and testing negative independent events?

e. No. P ( N ) = 0.85; P ( N $|$ B ) = 0.02. So, P ( N $|$ B ) does not equal P ( N ).

f. Are having breast cancer and testing negative mutually exclusive?

f. No. P ( B $\cap$ N ) = 0.0029. For B and N to be mutually exclusive, P ( B $\cap$ N ) must be zero.

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