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Question
How many times must a fair coin be tossed so that the probability of getting at least one head is more than 80%?
Solution
Let p denotes the probability of getting heads.
Let q denotes the probability of getting tails.
p=1/2
q=1-1/2=1/2
Suppose the coin is tossed n times.
Let X denote the number of times of getting heads in n trials.
`P(X=r)=""^nC_rp^rq^(n-r)=""^nC_r(1/2)^r(1/2)^(n-r)=""^nC_r(1/2)^n,r=0,1,2,3,4,......,n`
`P(X>=1)>80/100`
`=>P(X=1)+P(X=2)+.....+P(X=n)>80/100`
`=>P(X=1)+P(X=2)+.......+P(X=n+P(X=0))=P(X=0)>80/100`
`=>1-P(X=0)>80/100`
`=>P(X=0)<1/5`
`=>""^nC_0(1/2)^n<1/5`
`=>(1/2)^n<1/5`
`=>n=3,4,5.............`
So the fair coin should be tossed for 3 or more times for getting the required probability.
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