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Chapter 3: Problem 48

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?

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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).
04

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.
05

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.
06

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 \( \frac{2}{3} \).

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