Question

Adrian and Bobby play on opposing basketball teams. Bobby is always assigned to defend against Adrian. At the start of a game, Adrian feels confident with probability 3/4, and independently Bobby feels aggressive with probability 2/3. Adrian will take 20 shots at the basket if feeling confident, but only 12 if not. When Bobby is aggressive, Adrian scores on each shot with independent probability .4; otherwise Adrian’s probability of scoring is .6. Given that Adrian scored 10 times, what is the probability that Bobby was aggressive?

Answer 1

We are given here that:
P( confident Adrian ) = 3/4 and P( aggressive Bob) = 2/3

Also, we are given here that:
n (shots | Adrian confident) = 20 and n(shots | not confident Adrian) = 12

P(scores | Bob aggressive) = 0.4, P(scores | Bob not aggressive) = 0.6

We compute the probability of getting 10 scores in all the 4 cases first as:

Adrian	Bob	n	p	P(X = 10)
Confident	Aggressive	20	0.4
Confident	Not Aggressive	20	0.6
Not Confident	Aggressive	12	0.4
Not Confident	Not Aggressive	12	0.6

Therefore using law of total addition we get here :

= (3/4)(2/3)*0.1171 + (3/4)*(1/3)*0.1171 + (1/4)*(2/3)*0.0025 + (1/4)*(1/3)*0.0639

= 0.0936

Given that Adrian scored 10 times, probability that Bobby was aggressive is computed here as:

= [ (3/4)(2/3)*0.1171 + (1/4)*(2/3)*0.0025 ] / 0.0936

= 0.63

Therefore 0.63 is the required probability here.