A box contain $6$ coins, $3$ coins are regular, in two coins there are $"T"$ in both sides, and in one coin there is $"H"$ in both sides, we takig randomly one coin and throwing it $100$ times.
find the expectation value of the number of throwing that we got $"T"$
My try:
Let us denote $A_1$ the incident that we got a regular coin
$A_2$ the incident that we got the coin with $"T"$ in both sides
$A_3$ the incident that we got the coin with $"H"$ in both sides
$$E[X]=E[X|A_1]P(A_1)+E[X|A_2]P(A_2)+E[X|A_3]P(A_3)\\\frac 1 2E[X|A_1]\cdot \frac 1 3E[X|A_2]\cdot\frac 1 6\cdot E[X|A_3]$$
I'm stuck about calculating $E[X|A_i]$ for $1\leq i\leq 3$
Does a regular coin has "H" on one side and "T" on the other?
If so, than $E(X|A_1)= 50$