Problem 48 - A First Course in Probability

Question

Each of 2 cabinets identical in appearance has 2 drawers. Cabinet \(A\) contains a silver coin in each drawer, and cabinet \(B\) contains a silver coin in one of its drawers and a gold coin in the other. A cabinet is randomly selected, one of its drawers is opened, and a silver coin is found. What is the probability that there is a silver coin in the other drawer?

Step-by-Step Solution

Answer

The probability that there is a silver coin in the other drawer, given that a silver coin is found in one drawer, is \( \frac{2}{3} \).

Step 1: Identify the sample space and events

Let S be the sample space, A be the event that cabinet A is chosen containing two silver coins, and B be the event that cabinet B is chosen containing one silver coin and one gold coin. The sample space consists of the following possible outcomes: two silver coins in cabinet A, or one silver and one gold coin in cabinet B.

Step 2: Calculate the probability of each event

Both cabinets have an equal chance of being chosen, so the probability of selecting cabinet A (P(A)) and cabinet B (P(B)) is 1/2 for each.

Step 3: Calculate the conditional probabilities

We need the conditional probability of finding a silver coin in the other drawer given that we found a silver coin. This can be written as P(Silver in other drawer | Silver in one drawer). Using the conditional probability formula, this is equal to P(Silver in other drawer and Silver in one drawer) / P(Silver in one drawer).

Step 4: Calculate the numerator

The numerator is the probability of getting a silver coin in both drawers. Since cabinet A has two silver coins, the probability is P(A) * 1 = 1/2 because there is a 1/2 chance of selecting cabinet A, and the probability of getting a silver coin is 1.

Step 5: Calculate the denominator

The denominator is the probability of finding a silver coin in one drawer. Since cabinet A has two silver coins and cabinet B has one silver coin and one gold coin, the probability is P(A) * 1 + P(B) * 1/2 = (1/2) * 1 + (1/2) * (1/2) = 1/2 + 1/4 = 3/4. This is because there is a 1/2 chance of selecting cabinet A, with a 100% chance of getting a silver coin, and a 1/2 chance of selecting cabinet B with a 50% chance of getting a silver coin.

Step 6: Calculate the conditional probability

Now we can find the conditional probability P(Silver in other drawer | Silver in one drawer) = P(Silver in other drawer and Silver in one drawer) / P(Silver in one drawer) = (1/2) / (3/4) = (1/2) * (4/3) = 2/3. So, the probability that there is a silver coin in the other drawer is \( \frac{2}{3} \).