Understanding the Coin Purse Problem Setup

We have two purses, each containing a different mix of copper and silver coins. The task is to find the probability of picking a copper coin after first randomly selecting one of the purses.

Analyzing Coin Probabilities in Each Purse

Let's break down the contents and probabilities for each purse:

• Purse 1 Details:
  • Number of copper coins = 4
  • Number of silver coins = 3
  • Total number of coins = $4 + 3 = 7$
  • Probability of drawing a copper coin from Purse 1: $P(\text{Copper} | \text{Purse 1}) = \frac{4}{7}$
• Purse 2 Details:
  • Number of copper coins = 6
  • Number of silver coins = 4
  • Total number of coins = $6 + 4 = 10$
  • Probability of drawing a copper coin from Purse 2: $P(\text{Copper} | \text{Purse 2}) = \frac{6}{10} = \frac{3}{5}$

Applying the Law of Total Probability for Coin Selection

A purse is chosen randomly, meaning each purse has an equal chance of being selected:

• Probability of choosing Purse 1: $P(\text{Purse 1}) = \frac{1}{2}$
• Probability of choosing Purse 2: $P(\text{Purse 2}) = \frac{1}{2}$

We use the Law of Total Probability to find the overall probability of drawing a copper coin. This involves summing the probabilities of drawing a copper coin from each purse, weighted by the probability of selecting that purse.

The formula is:

Step-by-Step Calculation

1. Substitute the known probabilities into the formula: $P(\text{Copper}) = \left( \frac{4}{7} \right) \times \left( \frac{1}{2} \right) + \left( \frac{3}{5} \right) \times \left( \frac{1}{2} \right)$
2. Perform the multiplication for each term: $P(\text{Copper}) = \frac{4}{14} + \frac{3}{10}$
3. Simplify the first fraction: $P(\text{Copper}) = \frac{2}{7} + \frac{3}{10}$
4. Find a common denominator to add the fractions. The least common multiple of 7 and 10 is 70. $P(\text{Copper}) = \frac{2 \times 10}{7 \times 10} + \frac{3 \times 7}{10 \times 7}$
5. Add the fractions: $P(\text{Copper}) = \frac{20}{70} + \frac{21}{70} = \frac{20 + 21}{70}$
6. Final result: $P(\text{Copper}) = \frac{41}{70}$

Conclusion on Copper Coin Probability

The probability of drawing a copper coin after randomly choosing a purse is $\frac{41}{70}$.