Assume that a lost treasure will be in a certain area of the ocean with probability 0.4, and that a search of that area will find the treasure with probability 0.9 if it is there. What is the conditional probability of the treasure being in the area if the area is searched and no treasure is found?

the probability of the treasure being in the area given that no treasure is found is 0.0625 (6.25%)

Step-by-step explanation:

denoting the event N = no treasure is found, then

P (N) = probability that the lost treasure is in the area * probability that the lost treasure is not found in the treasure's area + probability that the lost treasure is not in the area * probability that the lost treasure is not found in other areas = 0.4*0.1 + 0.6*1 = 0.64

P (N) = 0.64

then we can get the conditional probability using the theorem of Bayes. Denoting the event A = the treasure being in the area

P (A/N) = P (A∩N) / P (N) = 0.4*0.1/0.64 = 0.0625 (6.25%)

where

P (A∩N) = probability of the treasure being in the area and no treasure is found

P (A/N) = probability of the treasure being in the area given that no treasure is found