A fair coin is flipped n times. Let E be the event "a head is obtained on the first flip" and let `F_(k)` be the event "exactly k heads are obtained".

Monjur Alahi

A fair coin is flipped n times. Let E be the event "a head is obtained on the first flip" and let `F_(k)` be the event "exactly k heads are obtained". Then the value of n/k for which E and `F_(k)` are independent is _____.


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Salim Ahmed Kawsar

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Correct Answer - 2
Here `P(E)=1/2and P(F_(k))=""^(n)C_(k).(1)/(2^(n))`
Also, `P(EnnF_(k))=p` (exactly k heads are obtained and head obtained in first filp)
`=1/2""^(n-1)C_(k-1)((1)/(2))^(n-1)`
Events E and `f_(k)` are independent. Therefore,
`P(EnnF_(k))=P(E).P(F_(k))`
`or ""^(n-1)C_(k-1)xx(1)/(2^(n))=1/2xx""^(n)C_(k)(1)/(2^(n))`
`or2xx""^(n-1)C_(k-1)=""^(n)C_(k)`
`orn=2k`

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