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Complement of an Event

An unbiased c...

Question

An unbiased coin is tossed $n$ times. Let $E_{1}$ be the event that both heads and tails are present in $n$ tosses. Let $E_{2}$ be the event that the coin shows up heads at most once in $n$ tosses. The value of $n$ for which $E_{1}$ and $E_{2}$ are independent is

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Solution

$n (S) = 2^{n}$
$n (E_{1}) = 2^{n} - 2$
$n (E_{2}) =^{n} C_{1} + 1 = n + 1 n (E_{1} \cap E_{2}) = n$
$∵ P (E_{1} \cap E_{2}) = P (E_{1}) \times P (E_{2}) \Rightarrow \frac{n}{2^{n}} = \frac{2^{n} - 2}{2^{n}} \times \frac{n + 1}{2^{n}} \Rightarrow n = 3$

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P (\frac{E_{i}}{A}) = \frac{P (E_{i}) P (\frac{A}{E_{i}})}{P (E_{1}) P (\frac{A}{E_{1}}) + P (E_{2}) P (\frac{A}{E_{2}})}, i = 1, 2

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