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?

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).
