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MA 485-585 PROBABILITY THEORY QUIZ #1 JANUARY 22 Time alotted: 15 min. Each question is 10 points. 1. We flip a fair coin 7 times. What is the conditional probability of the event that we get exactly 1 Head given the condition that the number of Heads is at most 3?
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Answer #1

let X is number of heads which follows binomial distribution with parameter n=7 and p=1/2

P(at most 3 heads)=P(X<=3)=P(X=0)+P(X=1)+P(X=2)+P(X=3)

=7C0(1/2)0(1/2)7+7C1(1/2)1(1/2)6+7C2(1/2)2(1/2)5+7C3(1/2)3(1/2)4

=0.0078+0.0547+0.1641+0.2734=0.5

hence P(1 heads given at most 3 heads)=P(X=1|X<=3)=P(X=1)/P(X<=3)=0.0547/0.5=0.1094

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