An unbiased coin is tossed 'n' times . Let the random variable X denote the number of times the head occurs . if P(X=1) ,P(X=2) and P(X=3) are in A.P , find the value of n.

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Please find below the solution to the asked query:

Probability of occurence of k heads in n tosses is:P1=12nnCkfor X=1 , P1=12nnC1=n2nfor X=2, P2=12nnC2=n!2!n-2!×2nfor X=3, P3=12nnC3=n!3!n-3!×2n2P2=P1+P3since they are in AP2×n!2!n-2!×2n=n2n+n!3!n-3!×2nn!n-2!×2n=n2n+nn-1n-2n-3!3!n-3!×2ncancelling n2n we get:nn-1n-2!n-2!=1+n-1n-23!=n-1=1+n-1n-23!=6n-1=6+n-1n-2=6n-6=6+n2-2n-n+2=9n-14=n2n2-9n+14=0n2-7n-2n+14=0nn-7-2n-7=0n-2n-7=0n=2 or n=7

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