What is the probability of getting tail on every even toss when we keep tossing a coin?

Question

For Bernoulli process, what is the probability of getting tail on every even toss when we keep tossing a coin? It is not same as getting n/2 tails in n tosses, right?

Answer 1

No, not the same. $n/2$ tails in $n$ tosses can be done in multiple ways, e.g. $TTTHHH$ in $6$ tries.

Compute the number of choices for the coin to fall on even tosses, and then on odd tosses, and multiply them.

Answer 2

Let $n>1$ be even. The number of total possibilities is $2^n$.

Number of desired ways (x T x T x T ...) is $2\times1\times2\times1\times2\times1 \cdots=2^{n/2}$, so the probability of desired outcome is $2^{-n/2}$.

For odd $n$, there are $2^{(n-1)/2}$ desired ways and the probability becomes $2^{-(n+1)/2}$.