Suppose we have two coins (coinA and coinB), and we conduct two independent experiments in which a single coin is tossed four times. The outcomes of the experiments are presented in the table below. We useH

to denote a Heads andT

to denote a Tails.

DenoteA:=P(H|A), i.e. the probability of gettingH when tossing coinA, andB:=P(H|B)

, i.e. the probability of gettingH when tossing coinB , as the parameters we want to estimate,x(i) as the outcome of each Experimenti,z(i) as the index (A orB) of the coin tossed in Experimenti.

Experiment | 1 | 2

Coin |B | A

Outcome | HTTH | HHHT

First assume we know from the table in each experiment which coin is tossed. Hence there is no "hidden" information and we can do the maximum likelihood estimation (MLE) directly.

1. As all the experiments are independent, the expression ofP(x(1),x(2)|A,B)should be in the form ofaA(1A)^b cB(1B)^d. Finda,b,c,d based on the outcomes in the table.