Box A contains 4 red and 6 blue balls. Box B contains 5 red and 10 blue balls. A box is selected at random and a ball is drawn. Find the probability that the ball is blue, given that it came from Box B.

To find the probability that the ball is blue, given that it came from Box B, we can use the concept of conditional probability.

Conditional probability is the probability of an event occurring given that another event has already occurred. In this case, the event is drawing a blue ball, and the condition is that the ball came from Box B.

Let's define the events:
A: Drawing a blue ball
B: The ball came from Box B

We are looking for P(A|B), which represents the probability of event A given event B. This can be calculated using the formula:

P(A|B) = P(A ∩ B) / P(B)

To find P(B), the probability that the ball came from Box B, we need to consider the relative number of boxes. Since there are two boxes (A and B) and they were selected at random, the probability of selecting Box B is 1/2.

Now, let's calculate P(A ∩ B), the probability that the ball is blue and came from Box B. Box B contains a total of 15 (5 red + 10 blue) balls, so the probability of drawing a blue ball from Box B is 10/15.

Putting it all together:
P(A|B) = P(A ∩ B) / P(B)
= (10/15) / (1/2)
= (10/15) * (2/1)
= (10/15) * (2/1)
= 20/15
= 4/3

Therefore, the probability that the ball is blue, given that it came from Box B, is 4/3 or approximately 0.1333 (to 4 decimal places).