Download Sample Exam A

EXAM PROBLEMS
(1) A school with 100 students oers French and Spanish. 20 students are
enrolled in French and 60 are enrolled in Spanish. 10 students are enrolled
in both French and Spanish. How many are enrolled in neither French nor
Spanish?
(2) You roll two dice. What is the probability that the largest number that
turns up is greater than 4?
(3) You select two numbers at random from the uniform distribution on [0, 1].
What is the probability that the product is greater than 1/2?
(4) You roll two dice. What is the probabililty that the sum of the two outcomes
is greater than 9 given that the maximum of the two outcomes is greater
than 5?
(5) Two players, A and B , start with three fair coins each. They ip coins and
compare the outcomes. If the coins match (two H or two T ) , player A gets
both coins. If they do not match, player B gets both coins. They continue
this process until one player has all of the coins.
(a) What is the sample space of the experiment that counts the number
of possible ips until one player has all of the coins?
(b) What is the probability that A has all of the coins after 4 ips?
(c) What is the probability that they both have 3 coins after two ips?
(6) Suppose X and Y are uniformly distributed on [0, 1] and are independent.
What is the probability that X + Y > 1/2 given that |X − Y | < 1/2?
(7) Two urns contain black and white marbles. The rst urn contains 10 black
and 3 white marbles. The second urn contains 7 black and 7 white marbles.
You randomly select one marble from each urn. What is the probability
that you end up with two black marbles?
(8) The time it takes for a light bulb to burn out is exponentially distributed
with mean of 3 years. What is the probability that a bulb will last 5 years
given that it lasts 4years? (The mean of the exponential distribution with
density function λ exp(λt) is 1/λ .)
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