Download Quiz 6 - Math Berkeley

Name:
Quiz 6
Problem 1. Determine if each of the following is true or false. No justification is required.
(a) A fair 6-sided dice is rolled three times. Let X be the random variable that is equal to twice the first roll minus
the sum of the second and third rolls. Then E(X) = 0.
(b) We flip a fair coin n times. The probability of seeing k heads is
k
n
1
.
2
k
(c) If A, B and C are events in a finite sample space such that
p(A ∩ B ∩ C) = p(A)p(B)p(C),
then A, B and C are mutually independent.
(d) If X and Y are random variables on a finite sample space, and V (X + Y ) = V (X) + V (Y ), then X and Y are
independent.
Problem 2. There’s an epidemic ravaging Mars! One common symptom of the disease is red eyes: 90% of infected
Martians have red eyes. But some Martians have red eyes naturally: 20% of uninfected Martians have red eyes. It
is estimated that 10% of Martians are infected.
(a) What percentage of Martians have red eyes?
(b) What percentage of Martians with red eyes are infected?
1
Problem 3. You go out gambling at a casino and decide to play the following game. You put some amount of
money into a pot and you guess either “heads” or “tails.” The casino matches your bet, so that the amount in the
pot is doubled. Then a fair coin is flipped. If your guess turns out correct, you win all of the money in the pot. If
your guess turns out incorrect, you lose the money in the pot to the casino.
You decide that your strategy for the night is to keep playing this game until you win, but at most 10 times.
On the first round, you will put $1 into the pot. Every time you lose, up to the 10th time, you will put in twice
the amount that you put into the pot the last time. How much money do you expect to gain tonight?
2