M362K Final Exam. May 10, 2002
Problem 1. Final Exams
The final exam for a certain math class is graded pass/fail. A randomly
chosen student from a probability class has a 40% chance of knowing the material
well. If he knows the material well, he has an 80% chance of passing the final.
If he doesn’t know the material well, he has a 40% chance of passing the final
anyway.
a) What is the probability of a randomly chosen student passing the final exam?
b) If a student passes, what is the probability that he knows the material?
Problem 2. Hot Pickups
There are 123,850 pickup trucks in Austin. Of these, 2,477 are stolen. [OK,
I’m making these numbers up.] Suppose that 100 randomly chosen pickups are
checked by the police.
a) What is the probability that exactly 3 of the 100 chosen pickups are stolen.
Give an EXACT answer, but do not evaluate it numerically.
b) Approximate this using the binomial distribution. What are nand p?
c) Approximate your answer to (b) using the Poisson distribution. What is λ?
Problem 3. Four Flushers
a) In poker, a “four-flush” is a hand with 4 cards of one suit, and one card
of another suit. (This sort of hand isn’t worth anything, and can be very
frustrating). Find the probability of a (5-card) hand being a four-flush.
b) A “three-flush” is a hand with 3 cards of one suit, one of another suit and one
of a third suit. (For instance, 3 spades, one heart and one club. A hand with 3
spades and 2 diamonds would not count.) Find the probability of a three-flush.
Problem 4. Joint distributions (2 pages)
Xand Yare continuous random variables with joint pdf
fX,Y (x, y) = n1/2 if 0 < x, 0 < y and x+y < 2
0 otherwise
a) Let Z=X+YFind the pdf of Z.
b) Find the pdf of X(that is, fX(x)) and that of Y.
c) Find the expectation and variance of X.
d) Find the covariance Cov(X, Y ).
Problem 5. Quidditch
A game of Quidditch lasts until a player catches the Golden Snitch, which
can happen at any time. Meanwhile, both teams try to score goals. The scoring
by the two teams is described by independent Poisson processes. The Gryffindor
team scores an average of 2.4 goals per hour, while the Slytherin team scores
an average of 2.0 goals per hour.
A certain match between Gryffindor and Slytherin lasts 20 hours.
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