probability question

Question

A bag has 8 green, 5 white and 7 purple balls. You decide to select two balls from the bag with replacement. Let Event A: Exactly one balls is green and Event B: at least one ball is white. a. Are events A and B mutually exclusive?b. What is the probability of selecting 2 green balls?

Answer 1

If you selected one green ball and one white ball, that would be both Event A and B. They are not mutually exclusive. Those 2 events can happen at the same time. To be mutually exclusive means if A then not B, and if B then not A.

If you're selecting one green ball then replacing it and selecting another green ball that seems to be probability of [8/(8+5+7)]² =8/20 = .4² = .16 You had a 40% chance each time. Just multiply the probabilities. 40% of 40% = 16% chance both are green.

But if the problem is interpreted as saying select 2 balls and then put them both back in the bag,replacing them. Then the chance of 2 green balls is just 40% x 7/20 or 40% x 35% = 14% That doesn't seem to be the problem though, as then "replacement" has no real application to the calculation. IF this is the solution and way the problem is interpreted, that's what you'd call a "trick question"

Answer 2

a. Events A and B are not mutually exclusive. Events A and B would be mutually exclusive if their probability of occurring together was zero. The probability of selecting a green ball on the first draw and a white ball on the second, one of the ways of having exactly one green ball and at least one white ball, is 8/20 x 5/20 = 40/400 = 0.10 > 0.

b. The probability of selecting 2 green balls is 8/20 x 8/20 = 64/400 = 0.16.

The key fact here is that when selecting balls with replacement, each selection is an independent event.