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