A purse contains 4 copper coins and 3 silver coins. A second purse contains 6 copper coins and 4 silver coins. A purse is chosen randomly and a coin is taken out of it. What is the probability that it is a copper coin?
41/70
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:
$P(\text{Copper}) = P(\text{Copper} | \text{Purse 1}) \times P(\text{Purse 1}) + P(\text{Copper} | \text{Purse 2}) \times P(\text{Purse 2})$Step-by-Step Calculation
- Substitute the known probabilities into the formula: $P(\text{Copper}) = \left( \frac{4}{7} \right) \times \left( \frac{1}{2} \right) + \left( \frac{3}{5} \right) \times \left( \frac{1}{2} \right)$
- Perform the multiplication for each term: $P(\text{Copper}) = \frac{4}{14} + \frac{3}{10}$
- Simplify the first fraction: $P(\text{Copper}) = \frac{2}{7} + \frac{3}{10}$
- Find a common denominator to add the fractions. The least common multiple of 7 and 10 is 70. $P(\text{Copper}) = \frac{2 \times 10}{7 \times 10} + \frac{3 \times 7}{10 \times 7}$
- Add the fractions: $P(\text{Copper}) = \frac{20}{70} + \frac{21}{70} = \frac{20 + 21}{70}$
- 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 $\frac{41}{70}$.
Important Questions from Types of Probability
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