selection of
colored balls from two bags. Assume that each bag contains 4 balls. Bag
a contains 2 red and 2 white, while bag b contains
2 red, 1 white, and 1 blue. You randomly select
one ball from bag a, note the color, and place the ball in bag b. You then
select a ball from bag b at random and make note of its color.
(1) What is the probability that both balls are red?
(2) What is the probability that both balls are red given
that the first ball you drew was red?
To solve these questions, we can use basic principles of probability.
(1) To find the probability that both balls are red, we need to consider all possible outcomes that satisfy this condition and divide it by the total number of possible outcomes.
The total number of possible outcomes for selecting one ball from each bag is 4 * 4 = 16, because each bag contains 4 balls.
Out of these 16 possible outcomes, we need to identify the outcomes where both balls are red. Since the first ball is selected from bag a, there are 2 red balls out of 4. Once the first red ball is selected and placed in bag b, there are now 3 balls in bag b. So, the probability of selecting another red ball from bag b is 2 out of 3.
Therefore, the probability of both balls being red is (2/4) * (2/3) = 4/12 = 1/3.
(2) To find the probability that both balls are red, given that the first ball drawn was red, we only need to consider the remaining outcomes after the first ball has been drawn.
Given that the first ball drawn was red, it means that the red ball from bag a was placed in bag b. So now, there are 5 balls in bag b (2 red, 1 white, and 2 blue).
The total number of possible outcomes after the first ball is drawn is 5.
Out of these 5 balls, 2 are red. Therefore, the probability of selecting another red ball from bag b is 2 out of 5.
Thus, the probability that both balls are red, given that the first ball drawn was red, is 2/5.