You and your friend have a fair coin each. Both of you toss the coins simultaneously, record the outcomes, and repeat the process, for a total of $n$ times. Let $X$ be the number of times you and your friend get the same outcome. Then $X$ is distributed as
SELECT ALL CORRECT OPTIONS
(a) $Binom(n,1/2)$
(b) $Geom(1/2)$
(c) $Binom(n,3/4)$
(d) $Binom(n,1/4)$
I thought the answer as $Binom(n,1/2)$ .. Since getting the same number of outcome is equiprobable. Please correct me if I am wrong.
Suppose that $X$ is random binomial variable. We have $X \sim Bin(n,p)$ which describes the number of successes $k$ in $n$ independent trials.
The question asks that both of toss the same outcome. The sample space for the tosses for both of you are given by
$$\Omega = \{ (H,H), (H,T) ,(T,H), (T,T)\} $$
the outcomes where they are the same is simply
$$E =\{ (H,H) ,(T,T)\} $$ the probability $p$ is $\frac{2}{4} =\frac{1}{2}$
Then we see that $X \sim Bin(n,\frac{1}{2})$. The geometric distribution is also constructed from independent bernoulli trials, however it is describing until you don't succeed.