A box contains 1 green ball, 2 blue balls, and 3 red balls. These balls are selected at random, one after another without replacement, and their colors are noted.
Find the conditional probability that the third ball is red given exactly one ball is red.
1/7
To find the conditional probability that the third ball is red given exactly one ball is red, we need to determine the probability of two scenarios:
1. Selecting a red ball as the first ball and a non-red ball as the second ball, followed by a red ball as the third ball.
2. Selecting a non-red ball as the first ball and a red ball as the second ball, followed by a red ball as the third ball.
Let's calculate each scenario step-by-step:
Scenario 1:
1. Probability of selecting a red ball as the first ball: There are a total of 6 balls (1 green, 2 blue, and 3 red), and 3 of them are red. Therefore, the probability of selecting a red ball as the first ball is 3/6.
2. Probability of selecting a non-red ball as the second ball: After selecting a red ball in the first step, there are now 5 balls left (1 green, 2 blue, and 2 red). Since we can't select the same ball twice, there are 2 non-red balls remaining. Therefore, the probability of selecting a non-red ball as the second ball is 2/5.
3. Probability of selecting a red ball as the third ball: After selecting a red ball in the first step and a non-red ball in the second step, there are now 4 balls left (1 green, 2 blue, and 1 red). Out of these, only 1 ball is red. Therefore, the probability of selecting a red ball as the third ball is 1/4.
Scenario 2:
1. Probability of selecting a non-red ball as the first ball: There are a total of 6 balls (1 green, 2 blue, and 3 red), and 3 of them are red. Therefore, the probability of selecting a non-red ball as the first ball is 3/6.
2. Probability of selecting a red ball as the second ball: After selecting a non-red ball in the first step, there are now 5 balls left (1 green, 2 blue, and 2 red). There are 2 red balls remaining. Therefore, the probability of selecting a red ball as the second ball is 2/5.
3. Probability of selecting a red ball as the third ball: After selecting a non-red ball in the first step and a red ball in the second step, there are now 4 balls left (1 green, 2 blue, and 1 red). Out of these, only 1 ball is red. Therefore, the probability of selecting a red ball as the third ball is 1/4.
To find the total probability, we add the probabilities of both scenarios: (3/6)*(2/5)*(1/4) + (3/6)*(2/5)*(1/4) = 2/20 + 2/20 = 4/20 = 1/5.
Therefore, the conditional probability that the third ball is red given exactly one ball is red is 1/5.