Suppose that we have a white urn containing five white balls and one red ball and we have a red urn containing one white ball and four red balls. An experiment consists of selecting at random a ball from the white urn and then (without replacing the first ball) selecting at random a ball from the urn having the color of the first ball. Find the probability that the second ball is red. What is the probability that the second ball is red is?
Mathematics · College · Wed Jan 13 2021
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Answer: To find the probability that the second ball selected is red, we need to consider two separate scenarios: (1) selecting a white ball from the white urn first and then selecting a red ball from the white urn, and (2) selecting a red ball from the white urn first and then selecting a red ball from the red urn. Here is the step-by-step calculation:
1. Probability of selecting a white ball from the white urn: There are 5 white balls and 1 red ball in the white urn, making a total of 6 balls. So, the probability of selecting a white ball from the white urn is 5/6.
2. Probability of then selecting a red ball from the white urn (after having chosen a white ball first): After removing one white ball, the white urn contains 4 white balls and 1 red ball. Therefore, the probability of picking a red ball from the white urn in the second draw is 1/5.
3. Combined probability for this scenario is the product of the probabilities from step 1 and step 2: (5/6) * (1/5) = 5/30.
Next, we consider the second scenario:
1. Probability of selecting a red ball from the white urn: There is only one red ball in the white urn out of a total of 6 balls. Therefore, the probability is 1/6.
2. Probability of then selecting a red ball from the red urn (after having chosen a red ball first): In the red urn, there is 1 white ball and 4 red balls, making a total of 5 balls. So, the probability of picking a red ball from the red urn is 4/5.
3. Combined probability for this scenario is the product of the probabilities from step 1 and step 2: (1/6) * (4/5) = 4/30.
Finally, to find the total probability that the second ball is red, we add the probabilities from each scenario:
Probability(second ball is red) = Probability(white first, then red) + Probability(red first, then red) = (5/30) + (4/30) = 9/30 = 3/10.
Therefore, the probability that the second ball selected is red is 3/10.