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How Many Times Must a Fair Coin Be Tossed So that the Probability of Getting at Least One Head is More than 80%? - Mathematics

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How many times must a fair coin be tossed so that the probability of getting at least one head is more than 80%?

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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.

Concept: Probability Examples and Solutions
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