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Introduction to Probability Name: MATH 3080 (Autumn 2009) Final Exam
Instructions:
- There are 105 points on the exam and an extra credit problem worth 10 points. 100 points counts as perfect, but, as always, points over 100 will help on your final grade.
- You may use your class textbook. No notes, calculators, computers, cell phones, con- sultants, or any other source material on the exam. Your cell phones must be turned off and may not be on the desk during the exam. You may not use your cell phone during the exam.
- You must show your work. No credit for correct answers alone.
- Simplify all answers unless explicitly asked not to. That is, your answers should be in terms of fractions (or in problem 2(b), a decimal) and there should be no binomial coefficients such as
μ 12 7
left.
Please do not write below this line.
Problem Points possible
Points earned
1 10
Extra Credit
Total
- (10 points) At a supermarket checkout, customers arrive at a rate of one every five minutes. What is the probability that at most 1 customer will arrive during a 10 minute period? (Hint: The number of arrivals X in 10 minutes is a Poisson random variable.) Do not attempt to simplify your answer.
- (5 points each) A fair coin is flipped independently 100 times. The random variable X counts the number of heads in these 100 flips.
(a) Write, but do not evaluate , a sum whose value is
P { 45 ≤ X ≤ 60 }.
(b) Using the DeMoivre-Laplace limit theorem, approximate P { 45 ≤ X ≤ 60 }.
(You will want to use the table on page 201 of the text. You need not use the continuity correction in this problem, just to make the computations easier.)
- (13 points)
- Urn 1 contains 3 White balls and 1 Black ball.
- Urn 2 contains 1 White and 4 Black balls.
One ball is drawn at random from urn 1 and transferred to urn 2. Then a ball is drawn at random from urn 2. It happens to be White. Find the probability that the transferred ball was White.
- (15 points) A room has three rows of 3 desks, as shown. (The 4 corner desks are marked with a • .)
Amy picks a seat at random. Then, Bill picks one of the remaining seats at random.
row 1 • • row 2 row 3 • • Let E be the event “Amy and Bill sit in the same row.” Let F be the event “Amy and Bill each sit in one of the four corner desks.” Are E and F independent events? (You must support your answer with computations!)
(d) (4 points) Find E [X] and var(X).
(e) (4 points) Find the density function fY (x) of the random variable Y = X^2.
(f) (5 points) The random variable X “picks” a point in (0, 4). Determine the ex- pected length of the piece that contains the point 1.
- In one play of a game, Amy rolls a fair die. If the die shows 1, Amy wins. If the die does not show 1, then Bill flips a fair coin. If the coin turns up Heads, Bill wins.
(a) (4 points) Find the probability that Amy wins and the probability that Bill wins in one play of the game.
(b) (6 points) The game is played repeatedly until either Amy or Bill has won. Find the probability that Amy wins the game.
- (10 points extra credit) A fair coin is flipped until 2 Heads in a row or 2 Tails in a row appear. The random variable N measures the flip on which the this occurs. So, for example, - N(THTHH) = 5 since we see 2 Heads in a row on the 5rd flip. - N(HTT) = 3 since we see 2 Tails in a row on the 3rd flip.
Find E[N].
