4.4 The Multiplication Rules and Conditional Probability
1. An urn contains 7 blue balls and 3 red balls. A ball is selected and its color noted.
Then it is replaced. A second ball is selected and its color noted. Find the probability
of selecting a blue ball and then a red ball.
2. An urn contains 7 blue balls and 3 red balls. A ball is selected, its color noted and it is
not replaced. A second ball is selected and its color noted. Find the probability
of selecting a blue ball and then a red ball.
3. In a class containing 12 men and 18 women, 2 students are selected at random to
give an impromptu speech. Find the probability that both are women.
4. If 18% of all Americans are underweight, find the probability that if three Americans
are selected at random, all will be underweight.
5. At a small college, the probability that a student takes physics and sociology is 0.092.
The probability that a student takes sociology is 0.73. find the probability that the
student is taking physics, given that he or she is taking sociology.
6. In a pizza restaurant, 95% of the customers order pizza. If 65% of the customers
order pizza and a salad, find the probability that a customer who orders pizza will also
order a salad.
7. In a statistics class there are 18 juniors and 10 seniors; 6 of the seniors are females,
and 12 of the juniors are males. If a student is selected at random, find the
probability that the student is a female given that the student is a junior.
8. The medal distribution from the 2000 summer Olympic Games is shown in the table:
United States
Russia
China
Australia
Others
Gold
39
32
28
16
186
Silver
25
28
16
25
205
Bronze
33
28
15
17
235
Choose one medal winner at random.
i)
Given that the winner was from the United States, find the probability
that the winner won the gold medal.
ii)
Find the probability that the winner was from the United States, given
that she or he won a gold medal.
iii)
Are the events ‘medal winner is from United States’ and ‘gold medal
was won’ independent? Explain your answer.
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9. At a local university 54.3% of incoming first-year students have computers. If three
students are selected at random, find the following probabilities:
i)
none have computers
ii)
at least one has a computer
iii)
all have computers
10. A lot of portable radios contains 15 good radios and 3 defective ones. If two are
selected and tested, find the probability that at least one will be defective.
11. If a die is rolled 3 times, find the probability of getting at least one even number.
12. A manufacturer makes two models of an item: model I, which accounts for 80% of
unit sales, and model II, which accounts for 20% of unit sales. Because of defects, the
manufacturer has to replace (or exchange) 10% of its model I and 18% of its model
II. If a model is selected at random, find the probability that it will be defective.
13. Urn 1 contains 5 red balls and 3 black balls. Urn 2 contains 3 red balls and 1 black
ball. Urn 3 contains 4 red balls and 2 black balls. If an urn is selected at random and
a ball is drawn, find the probability it will be red.
14. One white and one black die are rolled. Find the probability that the sum of their
numbers is 7 and that the number on the black die is larger than the number on the
white die.
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