Question

Suppose there are 1000 urns in a mathematician's warehouse. Each of these urns contains 20 marbles. 900 of the urns contain 15 green and 5 blue marbles, while 100 contain only blue marbles. Call these Type I and Type II urns, respectively. Suppose that we randomly select an urn and randomly draw a marble form it.

A. What is the probability that we selected a Type I urn, given that the ball we draw is blue? Given that the ball we draw is green?

B. Suppose I only catch a partial glimpse of the ball when it is drawn. This leads me to have 80% confidence that a blue ball was drawn. Assuming I update by Jeffrey Conditionalization, how confident should I then be that the ball was drawn from a Type I urn?

Answer 1

Given:

1000 urns -> Type I (900) with 15 green and 5 blue balls each | Type II (100) with 20 blue balls each

A) Given that a blue ball is drawn, probability that Type I urn is selected:- P(Type I urn|blue ball)

Using Bayes Theorem, we know that,

So, using the above theorem,

Now, P(Blue Ball) Can be written as (using total probability theorem

Thus,

Similarly,

Thus,   

We calculate P(Type I| Greenball) using the same method as above

P(Type I/Greenball) = = 1

1 or 100% probability is obvious here, as green is only present in Type I

B)

We know that P(Blue ball) = 0.8, P(Greenball) = 0.2

Using Jeffrey Conditionalization, we can say that

P(Type I) = P(Type I|Blueball) . P(Blueball) + P(Type I|Greenball) . P(Greenball)

=

=0.754

I am 75.4% confident that ball was drawn from Type I