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ECE 302 Division 1
Exam 1 Solutions, 2/17/2004.
- This is a closed-book exam. A formula sheet is provided. No calculators are allowed.
- Total number of points: 150. This exam counts for 25% of your final grade.
- You have 75 minutes to complete 4 problems.
- Be sure to fully and clearly explain all your answers.
- There will not be any discussion of grades. All re-grade requests must be submitted in writing, as stated in the course information handout.
Problem Points Score
1 40
Free points 5 5
TOTAL 150
Some random variables and their PMF’s:
Random variable PMF Mean Variance
Discrete uniform (^1) n , k = k 0 + 1, k 0 + 2,... , k 0 + n k 0 + n+1 2 n (^2) − 1 12 Geometric (1 − p)k−^1 p, k = 1, 2 , 3 ,... (^1) p p^12 − (^1) p
Binomial
n
k
(1^ −^ p)n−kpk,^ k^ = 0,^1 ,... , n^ pn^ np(1^ −^ p),
where (^) ( n k
n! (n − k)!k!
Problem 2 (50 points). An internet service provider (IAP) owns three servers. Each server has a 50% chance of being down, independently of the others.
a (10 points). Let X be the number of servers which are down at a particular time. Find pX (x), the probability mass function (PMF) of X. Solution. The state of each server is a Bernoulli trial with parameter 1/2, and therefore the number of servers which are down is a binomial random variable with parameter 1/2:
pX (x) =
k
1 2
if x = 0, 1 , 2 , 3 0 , otherwise
1 83 if^ x^ = 0,^3 8 if^ x^ = 1,^2 0 , otherwise
Joe is IAP’s customer who makes an attempt to access the internet every Sunday at noon (i.e., exactly one attempt is made every Sunday). He fails to access the internet if, at the time of his attempt, all three servers are down; otherwise, he succeeds. Assume that the states of the system on any number of different Sundays are independent.
b (10 points). What is the probability that Joe will fail to access the internet on his first attempt? Solution. If Joe failed to access the internet, all servers were down. The probability of this is, from Part a, 1/8.
c (10 points). What is the probability that Joe will be able to access the internet on exactly eight out of his first ten attempts? Solution. From Part b, Joe’s each attempt is a Bernoulli trial with probability of failure 1/8 and probability of success 7/8. Since the individual attempts are independent, the overall number of successes in ten attempts is a binomial random variable, and the probability of 8 successes is: ( 10 8
d (10 points). What is Joe’s expected number of failures to access the internet in the first 1000 attempts? Solution. The number of failures in the first 1000 attempts is a binomial random variable with parameter p = 1/8. Its expected value is np = 1000/8 = 125.
e (10 points). What is Joe’s expected number of attempts until his first failure to access the internet? (Include the failed attempt in your count of the expected number of attempts.) Solution. The number of attempts until and including the first failure is a geometric random variable with parameter p = 1/8. Therefore, the expected value is 1/p = 8.
Problem 3 (30 points). Al and Bill are running in the Democratic primaries. Their chances to win are 75% and 25%, respectively. John and George are running in the Republican primaries. Their chances to win are 40% and 60%, respectively. The outcome of the Democratic primaries and the outcome of the Republican primaries are independent. The winner of the Democratic primaries and the winner of the Republican primaries compete against each other in the general election for the President. If Al and John are competing in the general election, Al has 70% chance to win the general election, but if it’s Al against George then the chances are 50% for each candidate. If Bill gets through to the general election, he will have only 20% chance of winning against either one of the two Republican candidates.
a (10 points). Given that George won the Republican primaries, what is the conditional probability that George will win the general election? (Hint. To make your notation simple, define the following events: Gp = “George won his primaries” Gg = “George won the general election” Also, define similar events for other candidates.) Solution. Using the total probability theorem, we have:
P(Gg|Gp) = P(Gg ∩ Ap|Gp) + P(Gg ∩ Bp|Gp) = P(Gg|Ap ∩ Gp)P(Ap) + P(Gg|Bp ∩ Gp)P(Bp) =
b (10 points). What is the overall (unconditional) probability that George will become President? (Hint. Define G = “George became President,” and calculate P(G).) Solution. For G to occur, both Gp and Gg must occur:
P(G) = P(Gg ∩ Gp) = P(Gg|Gp)P(Gp) =
c (10 points). Given that George is President, what is the conditional probability that Al won the Democratic primaries? Solution. Using Bayes’ rule and the result from Part b,
P(Ap|G) =
P(G|Ap)P(Ap) P(G)
=
P(Gg ∩ Gp|Ap)P(Ap) P(G)
= P(Gg|Gp ∩ Ap)P(Gp|Ap)P(Ap) P(G)
=
d (5 points). Suppose Prof. Pollak decides to aim his first shot at Prof. Gelfand. Use your answers to Parts b and c, and apply the total probability theorem to find Prof. Pollak’s probability to win the duel. Solution. Let W be the event that Prof. Pollak wins the duel, let M be the event that he misses his first shot, and let H be the event that he hits Prof. Gelfand with his first shot. Then, from Part b, P(W |M ) = 0.3, and from Part c, P(W |H) = 3/13. Combining these through the total probability theorem, we get:
P(W ) = P(W |M )P(M ) + P(W |H)P(H) = 0. 3 · 0 .7 +
e (5 points). What is Prof. Pollak’s best strategy? Solution. First note that it does not make sense for Prof. Pollak to shoot at Prof. Shroff, because, if he succeeds, Prof. Gelfand will hit him with probability one. Thus, Prof. Pollak has two options for his first shot: (1) aim at Prof. Gelfand; (2) deliberately miss. Under Strategy (2), the probability of winning, from Part b, is 0.3. Under Strategy (1), the probability of winning, from Part d, is 0. 3 · (121/130) < 0 .3. In other words, Prof. Pollak’s chances are better under Strategy (2) than Strategy (1)! An interesting observation here is that the circumstances (in this case, the order in which the duelists shoot) are sometimes more important than the skills: even though Prof. Shroff is a better shot than Prof. Pollak, he has no chance in this duel.
