A 6-sided dice is rolled and M is the result obtained, then a coin is thrown M times and the number of heads obtained is N.

Question

My question is how to determine the conditional probability $\mathbb{P}(N=2|M=4)$, I'm trying to do it, but i didn't figure it out how to determine $\mathbb{P}(N=2 \cap M=4)$, and another question, how can I determine $\mathbb{P}(M=i)$, where $i$ is between $1$ and $6$? I thought that any probability $\mathbb{P}(M=i)$ would be $1/6$, but is it really this?

Thanks in advance, Fernando

Answer 1

If the dice is a regular one, then $\mathbb{P}(M=i)=1/6$, you are correct. Then, assuming $M$ is fixed, $N$ follows a binomial distribution of parameters $p=0.5$ (for a regular coin) and $n=M$.

So, for any $0\leq j\leq i$, $i\in [\![1,6]\!]$, $$\mathbb{P}(N=j | M=i) = \binom{i}{j}0.5^j\times 0.5^{i-j}$$