Problem 38 - A First Course in Probability

Question

Urn \(A\) has 5 white and 7 black balls. Urn \(B\) has 3 white and 12 black balls. We flip a fair coin. If the outcome is heads, then a ball from urn \(A\) is selected, whereas if the outcome is tails, then a ball from urn \(B\) is selected. Suppose that a white ball is selected. What is the probability that the coin landed tails?

Step-by-Step Solution

Answer

The probability that the coin landed tails, given that a white ball was selected, is \(\frac{12}{85}\).

Step 1: Probability of selecting a white ball from each urn

Let's first find the probability of selecting a white ball from each urn. We are given the number of white and black balls in each urn: Urn A: 5 white balls, 7 black balls (total 12 balls) Urn B: 3 white balls, 12 black balls (total 15 balls) The probability of selecting a white ball from each urn is: Urn A: \(P(W_A) = \frac{5}{12}\) Urn B: \(P(W_B) = \frac{3}{15} = \frac{1}{5}\)

Step 2: Probability of flipping a fair coin

The probability of flipping a fair coin to get heads (selecting a ball from urn A) or tails (selecting a ball from urn B) is equal: \(P(Heads) = P(A) = \frac{1}{2}\) \(P(Tails) = P(B) = \frac{1}{2}\)

Step 3: Use Bayes' theorem

Now that we have the probabilities of selecting a white ball from each urn and the probabilities of flipping a fair coin, we can use Bayes' theorem to calculate the probability of the coin landing tails given that a white ball was selected. Bayes’ theorem states: \(P(B|W) = \frac{P(W|B) P(B)}{P(W)}\) where \(P(B|W)\) = probability of the coin landing tails given that a white ball was selected, \(P(W|B)\) = probability of selecting a white ball from urn B, \(P(B)\) = probability of the coin landing tails, \(P(W)\) = probability of selecting a white ball from any urn.

Step 4: Calculate the probability of selecting a white ball from any urn

To calculate the probability of selecting a white ball from any urn, we can use the law of total probability: \(P(W) = P(W_A)P(A) + P(W_B)P(B)\) Plugging in the values we found in steps 1 and 2, this becomes: \(P(W) = \frac{5}{12} \times \frac{1}{2} + \frac{1}{5} \times \frac{1}{2} = \frac{5}{24} + \frac{1}{10}\) To add the fractions, we need to find a common denominator: \(P(W) = \frac{5}{24} + \frac{12}{24} = \frac{17}{24}\)

Step 5: Apply Bayes' theorem

Now we have all the values needed to apply Bayes' theorem: \(P(B|W) = \frac{P(W_B)P(B)}{P(W)} = \frac{(\frac{1}{5})(\frac{1}{2})}{\frac{17}{24}}\) To find the value of this fraction, we can first multiply the numerators together and then divide by the denominator: \(P(B|W) = \frac{\frac{1}{10}}{\frac{17}{24}}\) Now we can multiply by the reciprocal of the denominator: \(P(B|W) = \frac{1}{10} \times \frac{24}{17} = \frac{24}{170}\) Finally, we can simplify this fraction: \(P(B|W) = \frac{12}{85}\) So the probability that the coin landed tails (selecting a white ball from urn B) given that a white ball was selected is \(\frac{12}{85}\).