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Midterm

# 12W-Test2-Soln.pdf

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University of Waterloo

Statistics

STAT 230

Changbao Wu

Winter

Description

Short Solutions to Stat 230 Test 2, Feb 1, 2012
1. An irregular die is loaded to give the probabilities P(f1g) = 0:2, P(f2g) = 0:4, P(f3g) = P(f4g) =
P(f5g) = P(f6g) = 0:1. The die is rolled 8 times and results from di▯erent rolls are independent.
Find the probability that
[2] (a) A = \1 does not occur"
P(A) = (1 ▯ 0:2) = 0:8 = 0:1678
[1] (b) B = \2 does not occur"
P(B) = (1 ▯ 0:4) = 0:6 = 0:0168
[2] (c) C = \neither 1 nor 2 occurs"
P(C) = P(AB) = (1 ▯ 0:2 ▯ 0:4) = 0:4 = 0:00066
[2] (d) D = \both 1 and 2 occur"
P(D) = P(AB) = 1▯P(A[B) = 1▯[P(A)+P(B)▯P(AB)] = 1▯(0:8 +0:6 ▯0:4 ) = 0:8161 8 8 8 2. It is assumed that 0.05% of Canadian males are infected with HIV (i.e., HIV positive). A cheap blood
test for HIV has false negative rate at 2% and false positive rate at 0.5%. For each of the following
questions, de▯ne the events required for the calculation and simplify the results to the 5th
decimal point.
[2] (a) Find the probability that the test result for a randomly selected Canadian male is positive.
Let C = \A randomly selected Canadian male is HIV positive" and D = \The test result is
▯ ▯
positive". Then P(C) = 0:0005, P(D j C) = 0:02 and P(D j C) = 0:005.
▯ ▯
P(D) = P(C)P(D j C) + P(C)P(D j C) = 0:0005 ▯ (1 ▯ 0:02) + 0:9995 ▯ 0:005 = 0:00549
[2] (b) If the test result is positive, what’s the probability that the person is indeed HIV positve?
P(C)P(D j C) 0:0005 ▯ (1 ▯ 0:02)
P(C j D) = P(D) = 0:0054875 = 0:08929
[2] (c) If the test result is negative, what’s the probability that the person is indeed HIV negative (i.e.,
HIV-free)?
▯ ▯ ▯
P(C j D) = P(C)P(D j C) = 0:9995 ▯ (1 ▯ 0:005= 0:99999
P(D) 1 ▯ 0:0054875 3. Raptors and Lakers are scheduled to play a series of games in a third location (no home court
advantage for either team). The probability that Raptors win a game is 0.40 and results from

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