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An unbiased normal coin is tossed `n` times. Let `E_1:` event that both heads and tails are present `n` tosses. `E_2:` event that the coin shows up he

Monjur Alahi

An unbiased normal coin is tossed `n` times. Let `E_1:` event that both heads and tails are present `n` tosses. `E_2:` event that the coin shows up heads at most once. The value of `n` for which `E_1a n dE_2` are independent is _________.


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Salim Ahmed Kawsar

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`P(E_(1))=1-["P(all heads)+P(all tails")]`
`=1-[(1)/(2^(n))+(1)/(2^(n))]=1-(1)/(2^(n-1))`
`P(E_(2))=P("no head)+P(exactly one head")`
`=(1)/(2^(n))+^(""n)C_(1).(1)/(2^(n))=(n)/(2^(n))`
`P(E_(1)nnE_(2))=("exactly one head and"(n-1)till)`
`=^(""n)C_(1).(1)/(2).(1)/(2^(n-1))=(n)/(2^(n))`
If `E_(1)and E_(2)` are indipendent, then
`P(E_(1)nnE_(2))=P(E_(1)).P(E_(2))`
`implies(n)/(2^(n))=(1-(1)/(2^(n-1)))((n+1)/(2^(n)))`
`impliesn=(1-(1)/(2^(n-1)))(n+1)`

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