Multiple choice, please show your work, thank you!
Suppose that cars approaching an intersection turn left with probability 0.25, go straight with probability 0.45, or turn right with probability 0.3. We will assume that turns from car to car are independent. We define the random variables X, Y, and Z, to represent the number of cars in a sample of 50 cars that turn left, go straight, or turn right, respectively.
What is P(X+Y=30)?
A) 8.480%
B) 0.765%
C) 8.880%
D) 3.704%
Suppose that collisions at a busy intersection are assumed to occur randomly according to a Poisson process, at an average rate of 2.5 per week. For five one-week periods, what is the probability that there will be one week with no collisions, two weeks with exactly one collision, and two weeks with two or more collisions? You can assume that collisions are independent from week to week.
A) 0.0120
B) 0.3601
C) 0.0527
D) 0.0018
E-mail messages reach a server according to a Poisson process at an average rate of 36 messages per hour. The probability that any e-mail message will be SPAM is 1/3 independently of other messages. What is the probability of receiving at least one SPAM message in a 10-minute period?
A) 86.47%
B) 99.75%
C) 91.22%
D) 48.66%
Consider the Gambler's Ruin problem in which A and B both start with $5 and play a game with a series of independent rounds. The winner of any round gets $1 from the other player, and the game ends when one player has all of the money. Suppose that P(A wins any round) = 2/3 and P(B wins any round) = 1/3. What is the probability that A ends up with all the money, and thus has $10?
A) 66.67%
B) 86.83%
C) 96.97%
D) 73.66%
