To find the probability \( P(\text{penny} | \text{worth more than 10 cents}) \), we will use the formula for conditional probability:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
where:
- \( A \) is the event of choosing a penny,
- \( B \) is the event of choosing a coin worth more than 10 cents.
First, let's calculate the total number of coins in the jar:
\[ \text{Total coins} = 15 \text{ (pennies)} + 28 \text{ (nickels)} + 12 \text{ (dimes)} + 20 \text{ (quarters)} = 75 \text{ coins} \]
Next, we identify the coins that are worth more than 10 cents. The coins that fit this criterion are:
- Dimes (worth 10 cents, but not included in this consideration),
- Quarters (worth 25 cents).
So, the number of coins worth more than 10 cents is:
\[ \text{Nickels} = 0, \text{ Dimes} = 0, \text{ Quarters} = 20 \text{ (quarters)} \]
Thus, the total number of coins worth more than 10 cents is:
\[ \text{Total coins worth more than 10 cents} = 12 \text{ (dimes)} + 20 \text{ (quarters)} = 20 \text{ coins} \]
Now, we need to find \( P(A \cap B) \) which is the probability of choosing a penny AND a coin worth more than 10 cents. Since no pennies are worth more than 10 cents, this will be 0.
Therefore:
\[ P(A \cap B) = 0 \]
Next, we find \( P(B) \), which is the probability of choosing a coin that is worth more than 10 cents:
\[ P(B) = \frac{\text{Number of coins worth more than 10 cents}}{\text{Total number of coins}} = \frac{20}{75} \]
Now, we can calculate \( P(A | B) \):
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} = \frac{0}{\frac{20}{75}} = 0 \]
Thus the probability \( P(\text{penny} | \text{worth more than 10 cents}) \) is 0, which is not one of the provided answer choices:
- \( \frac{1}{3} \)
- \( \frac{1}{4} \)
- \( \frac{3}{11} \)
None of these are correct, as the final answer should be 0.
However, if we use the provided options, we can reasonably conclude that the correct interpretation should yield no applicable value of pennies existing under the condition set (value greater than 10 cents). Hence the final answer derived is:
0 (not an option). So likely not applicable from the provided choices.