Probability
(10 points) Question #1. Consider a random experiment where two fair six-sided dice are thrown.
(a) (5 points) Define the sample space of this experiment. How many outcomes are possible?
(b) (5 points) Show all the outcomes in the event ”the sum of both dice is 6”. What is the probability of this outcome?
(10 points) Question #2. A computer manufacturer receives 140 microchips from factory A, 160 microchips from factory B and 200 microchips from factory C. The chips have identical designs, but factory A produces chips with a failure rate of 0.003, while factory B produces chips with a failure rate of 0.002, and factory C produces chips with a failure rate of 0.009.
(a) (5 points) If the computer manufacturer chooses a chip at random, what is the probability that it fails?
(b) (5 points) What is the conditional probability that a chip selected at random was made by factory C, given that it failed?
(10 points) Question #3. Suppose an urn contains 5 red balls, 3 white balls, and 2 yellow balls. We draw a ball from the urn at random, write down its color and return the ball to urn.
We repeat this process infinitely.
(a) (2 points) What is the probability that we see the first white ball at the fifth draw?
(b) (2 points) What is the probability that we see no red balls in the first 4 draws?
(c) (3 points) What is the probability that we see 2 red balls in the first four draws?
(d) (3 points) What is the probability that we see exactly 3 white balls and 3 yellow balls in the first 10 draws?
(10 points) Question #4. Cars cross an intersection in the city at a rate of 5 per minute.
The number of cars passing the intersection X can be modeled as a Poisson random variable with parameter 5t:
P (X = k) = e−5t · (5t)k k!
(a) (5 points) What is the probability that only one car crosses the intersection after 1 minute? (You may write the answer in terms of e)
(b) (5 points) What is the probability that three or more cars cross the intersection (P [X ≥ 3]) as a function of t?
(10 points) Question #5. A random variable X has the probability density function: f (x) =
(cx3, x ∈ [0, 4] 0, elsewhere
(a) (5 points) Find the value of c.
(b) (5 points) Find the probability P [1 < X < 2].
(10 points) # Bonus Question. A coin with sides marked 0 and 1 is tossed an infinite number of times with the probability of 1 being p ∈ (0, 1). Let the resulting infinite sequence ω be parsed into blocks of length 10100. What is the probability that in infinitely many of these blocks, every term in the block of length 10100 is 1?
