Question

Suppose that a well-mixed urn contains 100 red and blue balls, in unknownproportion. Suppose you know that there are either 80 red balls or 20 red balls.You initially judge each of these possibilities to be equally likely and alwaysupdate your probabilities via Conditionalization when learning propositionalevidence.

1. Now suppose you can tell the 5 balls drawn are all of the same color and that you think the color is probably red but (due to the lighting)can't be sure. Your probability that they are all red goes to 90% whilethe probability that they are all blue does to 10%. What is your newprobability that urn contains 80 red balls?

Answer 1

1.

P(80 red | 5 balls are same with 90% chance of all red)

= 0.90*P(80 red | 5 red ) + 0.10*P(80 red | 5 blue )

= 0.90 * (80C5/100C5 * 0.5)/((80C5/100C5 * 0.5) + (20C5/100C5 * 0.5)) + 0.10* (20C5/100C5 * 0.5)/((20C5/100C5 * 0.5) + (80C5/100C5 * 0.5))

= 0.90 * ( 24040016 / 75287520 * 0.5)/(( 24040016 / 75287520 * 0.5) + ( 15504 /75287520* 0.5)) + 0.10* ( 15504 /75287520 * 0.5)/(( 15504 /75287520 * 0.5) + ( 24040016 /75287520 * 0.5))

= 0.8995 (answer)

(please UPVOTE)