What is the Maximum Likelihood Estimator for which coin was tossed? You are given three coins with the following probabilities of observing a Head when tossed:
Coin 1 has a P(H) = 1/2,
Coin 2 has a P(H) = 1/3,
Coin 3 has a P(H) = 1/4.
You observed two heads among three tosses, and you would like to find which coin the observations come from. What is the Maximum Likelihood Estimator for which coin was tossed (Coin 1, 2, or 3)?
I found this question on an online question site and I am confused of what the question is trying to ask.
I know that the possibility of getting 2 heads within 3 tosses is 3/8.
But I am not sure where to go from here. How do I solve this?
 A: Considering the Binomial experiment $\mathcal Bin(3,p)$  where two heads among three tosses are observed using one and only one of the three coins with different probabilities $p$, if $\eta$ denotes the unknown label of the coin $(\eta=1,2,3)$, the likelihood of the parameter $\eta$ associated with the available data $\mathcal D$ is
$$L(\eta|\mathcal D)=
\begin{cases}(1/2)(1/2)^{2} & \text{if }\eta=1\\
(2/3)(1/3)^2 & \text{if }\eta=2\\
(3/4)(1/4)^2 & \text{if }\eta=3\\
\end{cases}$$
$-$ where the multiplicative term$^\dagger$ $${3 \choose 2}=3$$ in the Binomial probability$$\mathbb P_p(X=2)={3 \choose 2}p^2(1-p)^1$$ is omitted since it does not depend on the parameter $\eta$ $-$ which corresponds to
$$L(\eta|\mathcal D)=
\begin{cases}0.125 & \text{if }\eta=1\\
0.074 & \text{if }\eta=2\\
0.047 & \text{if }\eta=3\\
\end{cases}$$
The maximum likelihood estimator is thus $\hat\eta=1$, meaning coin #1 is the most likely in producing this outcome.

$^\dagger$One should note that the likelihood function is usually defined up to a multiplicative constant.
A: If I understand things well now then one of the coins is thrown $3$ times and this resulted in $2$ heads and $1$ tail.
The question we ask ourselves is: which of the $3$ will be used most likely?
If $H$ denotes the number of heads and $C$ denotes the label of the coin then it is not difficult to find:

*

*$P(H=2\mid C=1)=3\times 2^{-3}$

*$P(H=2\mid C=2)=3\times(1/3)^2(2/3)$

*$P(H=2\mid C=3)=3\times(1/4)^2(3/4)$
That tells us that: $$P\left(H=2\mid C=1\right)=\max\left\{ P\left(H=2\mid C=\eta\right)\mid\eta\in\left\{ 1,2,3\right\} \right\} $$and makes us decide that most likely it was coin $1$ that was used.
Another expression for $P(H=2\mid C=\eta)$ is $L(C=\eta\mid H=2)$.
Here $P$ represents probability (of some event) and $L$ represents likelihood (of some parameter).
