Question

Suppose a game show called Bayesian Boxes has the contestant sample one marble from a box and guess what type of box it is. When the contestant picks a box, the chance of each box type for this round is as follows:

Type A: 75%, Type B: 20%, and Type C: 5%
The chance of drawing a red marble from Type A is 2%, from Type B is 25%, and from Type C is 98%.
The contestant selects a box and draws a marble. What is the chance of drawing a red marble? (Give the exact decimal answer.)
If the contestant does draw a red marble, what is the chance it came from box Type C? (Give the decimal answer rounded to at least four places.)

Answer 1

0.114

0.049

Explanation:

For the boxes:

Type A - 75% = 0.75

Type B - 20% = 0.2

Type C - 5% = 0.05

For the marbles in each box:

Type A - 2% = 0.02

Type B - 20% = 0.25

Type C - 98% = 0.98

Then you can multiply the chance of drawing each box by the chance of drawing a red marble from that box and add all of those together:

0.75 * 0.02 + 0.2 * 0.25 + 0.05 * 0.98

= 0.015 + 0.05 + 0.049

= 0.114

The specific chance for box Type C is 0.05 * 0.98 = 0.049