Question

Two urns each contain green balls and red balls. Urn I contains 10 green balls and 8 red balls, and Urn II contains 3 green balls and 10 red balls. A ball is drawn from each urn. Find P(red and red)

Answer 1

The probability of drawing a red ball from either urn is
P(red) = (number of red balls in urn)/(total balls in urn)

Use that to find the probability of drawing a red ball from the first and second urns.
Drawing a ball from the first urn does not affect drawing a ball from the other urn - the events are independent. So the probability of drawing two reds is:
P(red and red) = P(red from Urn I) x (red from Urn II)

Answer 2

is it 18/31?

Answer 3

Well, the probability from Urn I is 8/18, and the probability from Urn II is 10/13.
Multiplying together gives 80/234, which simplifies to 40/117.

Answer 4

You can't treat drawing two balls from the different urns as a single event: you have to consider the probability of drawing a ball from either urn.