- #1

LokLe

- 37

- 5

- Homework Statement
- There are three boxes:

Box 1 contains one white ball and one black one.

Box 2 contains two white balls and one black one.

Box 3 contains three white balls and one black one.

Suppose I pick one box, then I pick a ball at random from this box and I show you the ball. Suppose the ball is white.

Which box would you guess it came from? What is your probability of guessing right?

- Relevant Equations
- P(H|E) = P(H)P(E|H)/P(E)

I calculated the probability of box 1/2/3 given white.

The probabilities summed up are not equal to 1. Is this normal or are there mistakes in my calculation?

Thank you!

*P(Box 1 | white)*

= P(Box 1)*P(White | Box 1)/(P(Box 1)*P(White | Box 1) + P(Not Box 1)*P(White | Not Box 1))

= ((1/3)*(1/2)) / (((1/3)*(1/2) + (2/3)*(5/7)

= 7/27

P(Box 2 | white)

= P(Box 2)*P(White | Box 2)/(P(Box 2)*P(White | Box 2) + P(Not Box 2)*P(White | Not Box 2))

= ((1/3)*(2/3)) / ((1/3)*(2/3) + (2/3)*(4/6))

= 1/3

P(Box 3 | white) = P(Box 3)*P(White | Box 3)/(P(Box 3)*P(White | Box 3) + P(Not Box 3)*P(White | Not Box 3)) =

= ((1/3)*(3/4)) / ((1/3)*(3/4) + (2/3)*(3/5))

= 5/13

Therefore, you should guess box 3.= P(Box 1)*P(White | Box 1)/(P(Box 1)*P(White | Box 1) + P(Not Box 1)*P(White | Not Box 1))

= ((1/3)*(1/2)) / (((1/3)*(1/2) + (2/3)*(5/7)

= 7/27

P(Box 2 | white)

= P(Box 2)*P(White | Box 2)/(P(Box 2)*P(White | Box 2) + P(Not Box 2)*P(White | Not Box 2))

= ((1/3)*(2/3)) / ((1/3)*(2/3) + (2/3)*(4/6))

= 1/3

P(Box 3 | white) = P(Box 3)*P(White | Box 3)/(P(Box 3)*P(White | Box 3) + P(Not Box 3)*P(White | Not Box 3)) =

= ((1/3)*(3/4)) / ((1/3)*(3/4) + (2/3)*(3/5))

= 5/13

Therefore, you should guess box 3.

The probabilities summed up are not equal to 1. Is this normal or are there mistakes in my calculation?

Thank you!