STAT 230
Test 2
May 31, 2006
3:30 – 4:10 pm
Name:
ID:
UWUserid:
Circle your section
: Section 01 (1:30pm)
Section 02 (11:30am)
Section 03 (9:30am)
1. In the ﬁnal stage of a game show, two contestants take it in turns to answer questions.
Each question is picked at random from a large stack, of which 40% relate to sports and the
remaining 60% are about history. The ﬁrst contestant, Angela, has a 70% chance of answering
a sports question correctly, and a 50% chance of answering a history question correctly. The
second contestant, Brian, has a 20% chance of answering a sports question correctly, and a
90% chance of answering a history question correctly. If a contestant fails to answer a question
correctly, the question is discarded and a new question picked to ask the other contestant. The
cycle continues until one of the contestants answers correctly. The ﬁrst question is directed at
Angela, and the ﬁrst contestant to answer a question correctly wins the game. Assume that
contestants’ answers to diﬀerent questions are independent.
[2] (a) Show that at each turn, Angela has a probability of 0.58 of answering her question correctly.
P
(correct) =
P
(correct

sports)
P
(sports) +
P
(correct

history)
P
(history)
= 0
.
4
×
0
.
7 + 0
.
6
×
0
.
5 = 0
.
58
[2] (b) Show that at each turn, Brian has a probability of 0.62 of answering his question correctly
P
(correct) =
P
(correct

sports)
P
(sports) +
P
(correct

history)
P
(history)
= 0
.
4
×
0
.
2 + 0
.
6
×
0
.
9 = 0
.
62
[2] (c) Let round 1 begin when Angela is asked her ﬁrst question, and round 2 when she is asked
her second question. In general, round
n
begins when Angela is asked her
n
th
question,
and ends before she is asked her (
n
+ 1)
th
question. Find the probability that Angela wins
on round 3.
P
(A wins on round 3) =
P
(both answer wrong on round 1)
×
P
(both wrong on round 2)
×
P
(A right on round 3)
= (0
.
42
×
0
.
38)
2
×
0
.
58 = 0
.
01477
[2] (d) Find the probability that Brian wins on round 3
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(B wins on round 3) =
P
(both answer wrong on round 1)
×
P
(both wrong on round 2)
×
P
(A wrong on round 3)
×
P
(B right on round 3)
= (0
.
42
×
0
.
38)
2
×
0
.
42
×
0
.
62 = 0
.
00663
[2] (e) Find the probability that the game ends on round 3.
P
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 Probability, Wilma, Fred Flintstone

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