Question 1: On the first page of your assignment, write your name and student number.
• OLG chooses a winning number w in S = {0, 1, 2, . . . , 999}.
• If John wants to play, he pays $1 and chooses a number x in S.
If x = w, then John receives $700 from OLG. In this case, John wins $699.
Otherwise, x 6= w and John does not receive anything. In this case, John loses $1.
Assume that
• John plays this game once per day for one year (i.e., for 365 days),
• each day, OLG chooses a new winning number,
• each day, John chooses x uniformly at random from the set S, independently from
previous choices.
Define the random variable X to be the total amount of dollars that John wins during one year. Determine the expected value E(X). (Hint: Use Linearity of Expectation.)
Question 3: The Ottawa Senators and the Toronto Maple Leafs play a best-of-seven series: These two hockey teams play against each other until one of them has won four games. Assume that
• in any game, the Sens have a probability of 3/4 of defeating the Leafs, • the results of the games are independent.
Determine the probability that seven games are played in this series.
Question 4: Let X1, X2, . . . , Xn be a sequence of mutually independent random variables.
For each i with 1 ≤ i ≤ n,
• the variable Xi
is equal to either 0 or n + 1,
• E(Xi) = 1.
Determine
Pr(X1 + X2 + · · · + Xn ≤ n).