Description
1. If you have two standard six-sided dice, each with uniform probability of
landing on each counting number from 1 to 6. What is the probability of
rolling doubles (both dice landing on the same number)?
2. Let X and Y be two independent random variables. P[X,Y]= 0.2 and P[X] =0.5.
Find P[Y].
3. A drunk person is walking on the road. With probability 0.6 he takes a step
forward and with probability 0.4 he takes a step backward. After 10 steps,
what is the probability that he is at his starting position? Just the expression
is sufficient.
4. Let X, Y and Z be three random variables. E[X]= 2, Var(X) =1 and E[Y]=3. X
and Y are independent of each other. Z = X2 Y. Find E[Z].
5. Find the mean, median and variance of the following numbers. 1, 6, -1, 4, 10.
6. A gambler bets n times. Each time the gambler bets, 20% of the time he wins
$10 and 80% of the time he loses $5. What is expected gain (which can be
negative) after n bets?
7. You are drawing cards from a deck (consisting of the standard 52 cards) one
at a time without replacement. Let X and Y denote the first and second cards
you draw from the deck. You observe that the first card is a spade. What is
the probability that the second card you draw is also a spade?
8. Consider two urns. The first contains two white and seven black balls, and
the second contains five white and six black balls. We flip a fair coin and then
draw a ball from the first urn or the second urn depending on whether the
outcome was heads or tails. What is the conditional probability that the
outcome of the toss was heads given that a white ball was selected?
9. Suppose you toss a coin 10 times. The probability of getting a head in each
toss is p. What is the probability that you get more than 6 heads in the 10
coin tosses. Just the expression is sufficient.
10. A man enters a betting competition. Each time the man bets, the probability
of winning is p. What is the probability that he wins for the first time after n
bets? Name this distribution.
