Solving a Probability Problem with Two Urns and a Coin Flip

Urn 1 has five white and seven black balls. Urn 2 has three white and twelve black balls. We flip a fair coin. If the outcome is heads, then a ball from urn 1 is selected, while if the outcome is tails, then a ball from urn 2 is selected. Suppose that a white ball is selected. What is the probability that the coin landed tails?

Let's begin by identifying the possible outcomes. We could pick a white ball from urn 1 after a heads outcome or a white ball from urn 2 after a tails outcome. Since we have only one white ball from urn 1 and three white balls from urn 2, we know that the probability of selecting a white ball from urn 1 is:

P(white ball from urn 1) = 1 / (5 + 7) = 1 / 12

The probability of selecting a white ball from urn 2 is:

P(white ball from urn 2) = 3 / (3 + 12) = 1 / 5

Now, we need to find the probability of the coin landing tails, given that we have selected a white ball. We can use Bayes' Theorem to calculate this probability:

P(tails | white ball) = P(white ball | tails) * P(tails) / P(white ball)

P(white ball | tails) is simply the probability of selecting a white ball from urn 2, which we have already calculated as 1/5. P(tails) is the probability of the coin landing tails, which is 1/2 since the coin is fair. Finally, P(white ball) is the probability of selecting a white ball from either urn, which we can calculate as follows:

P(white ball) = P(white ball from urn 1) * P(heads) + P(white ball from urn 2) * P(tails)

P(white ball) = (1/12) * (1/2) + (3/15) * (1/2) = 1/6

Substituting these values into Bayes' Theorem, we get:

P(tails | white ball) = (1/5) * (1/2) / (1/6) = 3/10

Therefore, the probability that the coin landed tails, given that we have selected a white ball, is 3/10.

Solving probability problems involving urns and coin flips can be tricky, but by breaking down the problem into individual probabilities and using Bayes' Theorem, we can arrive at the correct answer. In this case, the probability that the coin landed tails, given that we have selected a white ball, is 3/10.