With probability 0.6, the present was hidden by mon; with probability 0.4, it was hidden by dad. When mom hidesthe present, she hides it upstairs 70 percent of the time anddownstairs 30 percent of the time. Dad is equally likely tohide it upstairs r downstairs.

a) What is the probability that the present is upstairs?

b) Given that it is downstairs, what is the probabilityit was hidden by dad?

a) the probability that the present is upstairs is 0.62 (62%)

b) the probability that it was hidden by dad, given that it is downstairs is 0.526 (52.6%)

Step-by-step explanation:

denoting the event U=the present is upstairs, then the probability P (U) is

P (U) = probability that was hidden by mum * probability that she hide it upstairs + probability that was hidden by dad * probability that he hide it upstairs = 0.6 * 0.7 + 0.4 * 0.5 = 0.62

P (U) = 0.62

defining the event S = the present is downstairs, then

P (S) = 1-P (U) = 1 - 0.62 = 0.38

for b) we use the theorem of Bayes. Denoting the event D = the present was hidden by dad

P (D/S) = P (D∩S) / P (S) = 0.4*0.5/0.38 = 0.526

P (D/U) = 0.526

where

P (D∩S) = probability that the present was hidden by dad and it is also is downstairs

P (D/S) = probability that the present was hidden by dad, given that it is downstairs