45-733 PROBABILITY AND STATISTICS I

Practice Conditional Probability Problems From Past Exams and Class Examples

- Suppose we have three urns. Urn I has 50 red balls, 40 white,
and 10 green. Urn II has 20 red balls, 60 white, and 20 green. Urn III
has 30 red balls, 30 white, and 40 green. We draw a card randomly from a
deck of 52 cards. If the card drawn is a spade we randomly draw 4 balls
from Urn I. If the card drawn is a heart or a diamond, we randomly draw
4 balls from Urn II. If the card drawn is a club we randomly draw 4 balls
from Urn III. If 2 green and 2 white balls were drawn, what is the
probability they came from Urn I?

Answer - Suppose we have two urns. Urn I contains 12 green balls,
16 yellow, and 2 red. Urn II contains 6 green balls, 5 yellow, and 4 red.
We flip a coin. If a head we draw one ball randomly from Urn I, if a tail
we draw one ball randomly from Urn II. Given that a yellow ball is drawn,
what is the probability that it came from Urn II. Given that a red ball is
drawn, what is the probability it came from Urn I?

Answer - A high ranking U.S. official feels that the probability is .3
that a particular country will devalue its currency in the next six months.
A news report is then received that the devaluation will occur. In the
past, such news reports have been received before 70 percent of the actual
devaluations and before 25 percent of rumored devaluations that did not,
in fact, occur. Given the information in the news report, calculate the
probability the official should now assign to the likelihood of a
devaluation?

Answer - A die is rolled 3 times. If it is know that face 1 appeared
at least once, what is the probability that it appeared exactly once?

Answer - An urn contains 6 red and 4 white balls. Three balls are
removed from the urn. Find the probability that all 3 of the removed
balls are red if it is known that at least 1 of the removed balls is
red?

- We have a new test for detecting a disease. One person in
every 100,000 is known to have the disease. If a person has the disease
the probability of a positive test is .95. If a person does not have the
disease the probability of a positive test is .10. We randomly draw an
individual from the population and administer the test. If the test is
negative, what is the probability the person does not have the
disease?

- Suppose we have four chests each having two drawers. Chest
1 has a gold coin in each drawer, chest 2 has a silver coin in each drawer,
chest 3 has a gold coin in one drawer and the other drawer is empty, and
chest 4 has a silver coin in one drawer and a gold coin in the other
drawer. A chest is selected at random and a drawer is opened. In the
drawer is a gold coin. What is the probability that the other drawer
has a silver coin in it?

- A woman fires 20 shots independently at a target. The
probability is .9 that she will hit the target on any given shot. What
is the probability that she has hit the target at least twice if it is
known that she has hit the target at least once?