[Bayes' Theorem] Finding the probability of guessing correctly

[LokLe]

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.

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.

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

Thank you!

 

[Orodruin]

There are mistakes in your calculation.

How do you compute P(white)?

 

[LokLe]

I did not calculate P(white). Instead, I calculated P(Box (1/2/3) | white) + P(Not Box (1/2/3) | Not white)

 

[LokLe]

I did not calculate P(white). Instead, I calculated P(Box (1/2/3) | white) + P(Not Box (1/2/3) | Not white)

P(white) = P (total number of white balls/total number of balls) = 6/9 = 2/3

 

[Orodruin]

Are all balls drawn with equal probability?

 

[LokLe]

Are all balls drawn with equal probability?

No. In box 3, the probability of drawing a white ball is higher = 3/4.

In other boxes:
P(box 1) = 1/2
P(box 2) = 2/3

 

[Orodruin]

No. In box 3, the probability of drawing a white ball is higher = 3/4.

In other boxes:
P(box 1) = 1/2
P(box 2) = 2/3

Not the probability of drawing a white ball, the probability of drawing any specific ball. For example, how do you compute P(white|not box 1)?

 

[PeroK]

What happens on average if you repeat the process $36$ times? That should give you the probability of drawing each ball.

And, in fact, all the data you need.

 

[LokLe]

What happens on average if you repeat the process $36$ times? That should give you the probability of drawing each ball.

So I drew a tree diagram and found P(White | Not box (1/2/3)).
P(White | Not box 1) = 1/3*2/3 + 1/3*3/4 = 17/36
P(White | Not box 2) = 1/3*1/2 + 1/3*3/4 = 15/36
P(White | Not box 3) = 1/3*1/2 + 1/3*2/3 = 14/36

So by using these values:
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 + 17/36)
= 6/23

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 + 15/36)
= 8/23

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 + 14/36)
= 9/23

The answer is box 3.

This should be the correct answer. Thank you all for the help!

 

[PeroK]

From the 36 experiments:

Box 1 and white = 6
Box 2 and white = 8
Box 3 and white = 9

That gives you 23/36 as the probability of drawing a white ball. And the conditional probability of Box 1 given white as 6/23 etc.

 

[LokLe]

From the 36 experiments:

Box 1 and white = 6
Box 2 and white = 8
Box 3 and white = 9

That gives you 23/36 as the probability of drawing a white ball. And the conditional probability of Box 1 given white as 6/23 etc.

How did you calculate, for example, box 1 and white?

 

[LokLe]

How did you calculate, for example, box 1 and white?

Oh It is 1/2*1/3 = 1/6 = 6/36.

 

[Orodruin]

From the 36 experiments:

Box 1 and white = 6
Box 2 and white = 8
Box 3 and white = 9

That gives you 23/36 as the probability of drawing a white ball. And the conditional probability of Box 1 given white as 6/23 etc.

The main question is though: Do you understand why your original approach failed?

Edit: That was clearly intended to quote one of OP’s posts, not @PeroK ...

 

[LokLe]

The main question is though: Do you understand why your original approach failed?

My original P(White | box (1/2/3)) is not correct since it is not just about calculating the number of white balls in other boxes. The whole case should be considered.