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?