Suppose we have two coins (coin A and coin B ), 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 use H to denote a Heads and T to denote a Tails. Denote θA:=P(H|A) , i.e. the probability of getting H when tossing coin A , and θB:=P(H|B) , i.e. the probability of getting H when tossing coin B , as the parameters we want to estimate, x(i) as the outcome of each Experiment i , z(i) as the index ( A or B ) of the coin tossed in Experiment i .

Question

Suppose we have two coins (coin  A  and coin  B ), 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 use  H  to denote a Heads and  T  to denote a Tails. Denote  θA:=P(H|A) , i.e. the probability of getting  H  when tossing coin  A , and  θB:=P(H|B) , i.e. the probability of getting  H  when tossing coin  B , as the parameters we want to estimate,  x(i)  as the outcome of each Experiment  i ,  z(i)  as the index ( A  or  B ) of the coin tossed in Experiment  i .

Experiment Coin	       Outcome
1	             B	                 H T T H
2	             A	                  H H H T
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.

As all the experiments are independent, the expression of  P(x(1),x(2)|θA,θB)  should be in the form of  θaA(1−θA)bθcB(1−θB)d . Find  a,b,c,d  based on the outcomes in the table.

a=  
 unanswered 
b=  
 unanswered 
c=  
 unanswered 
d=