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(IELM151)[2010](s)midterm~3541^_76482.pdf
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IELM 151 Engineering Probability and Statistics Spring 2010
March 26, Friday Jiheng Zhang

Midterm
Name: Student No.: No notes, no books, and no calculators are allowed. Only a one side A4 cheat sheet is allowed. There are 6 problems. The total score is 30 points. Please show major steps in analysis in order to get partial credits. The exam time is 70 minutes.
1 2 3 4 5 6 Total


1. (5 points) A gambler has in his pocket a fair coin and a two-head coin (the probability of showing a head is 1). He selects one of the coins at random.
(a)
If a head is shown after one .ip, what is the probability that the coin is fair?

(b)
If an other head is shown after the second .ip (that is you see two heads in the .rst two .ips), what is the probability that the coin is fair?


Solution:
(a)
P (H|fair)P (fair) (1/2)(1/2) 1
P (fair|H)= ==
P (H|fair)P (fair)+ P (H|unfair)P (unfair) (1/2)(1/2) + (1)(1/2) 3
(b)
P (HH|fair)P (fair) (1/4)(1/2)
P (fair|HH)= =
P (HH|fair)P (fair)+ P (HH|unfair)P (unfair) (1/4)(1/2) + (1)(1/2)
1
=
5
2. (5 points) Suppose Tony and Paul are in a team of 10 players. Suppose 6 of them will be randomly chosen to play a game.
(a)
What is the probability that Tony will be chosen?

(b)
What is the probability that both Tony and Paul will be chosen?

(c)
Given Paul is chosen, what is the probability that Tony will be chosen?


(d) Let X = . 1 0 if Tony is chosen if Tony is NOT chosen
. 1 if Paul is chosen
Y = 0 if Paul is NOT chosen

Are X and Y independent? If Yes, prove it. If not, calculate the covariance between X and Y .
Solution:
6
(a) The probability that Tony is selected is
10 65 1
(b) The probability that both will be chosen is =
109 3
(c) By Bayes formula, the probability is 5
9
66
(d) By the result of (a), P (X = 1) = ,P (Y =1)= .
10 10
1
By the result of (b), P (X =1,Y = 1) = .= P (X = 1)P (Y = 1). So X, Y are not
3
independent.
Cov(X, Y )= E(XY ) . E(X)E(Y )= P (X =1,Y = 1) . P (X = 1)P (Y = 1) = . 2

75 .
The negative covariance means that if Tony is chosen, it is less likely Paul will be chosen.
Remark:
(d) is essentially the same as the lower left slide on Page 4 of Notes 0n Chapter 4b.
3. (4 points) Tony and Paul each has a fair die. After one toss, if Tonys die shows a number bigger than Pauls, then Tony wins; if Tonys die shows a number less than Pauls, then Paul wins; if two dies show the same number, they continue to the next round, following the same rule.
(a)
What is the probability Tony wins?

(b)
Given that the game ends in the .rst round, what is the probability that Tony wins.

(c)
Given that Tony wins, what is the probability that he wins in the second round?

(d)
What is the probability that the game will .nishes at the 3rd round?


Solution:
Let T denote the event that Tony wins and X denote the number of total rounds.
(a) Since Tony and Paul are equally