To find the probability that the first coin is a nickel and the second coin is a quarter, let's break down the problem step-by-step.
1. Probability that the first coin is a nickel:
- Karin has 13 coins in total.
- Out of these, 4 coins are nickels.
- The probability that the first coin drawn is a nickel is calculated as follows:
[tex]\[
P(\text{First coin is a nickel}) = \frac{\text{Number of nickels}}{\text{Total number of coins}} = \frac{4}{13} \approx 0.3077
\][/tex]
2. Probability that the second coin is a quarter given that the first coin was a nickel:
- If the first coin drawn is a nickel, Karin will then have 12 coins left.
- Out of these 12 coins, we know that there are still 2 quarters (since choosing a nickel doesn't affect the number of quarters).
- The probability that the second coin drawn is a quarter, given that the first coin was a nickel, is:
[tex]\[
P(\text{Second coin is a quarter | First coin is a nickel}) = \frac{\text{Number of quarters}}{\text{Remaining number of coins}} = \frac{2}{12} = \frac{1}{6} \approx 0.1667
\][/tex]
3. Combined probability:
- The combined probability of both events happening (i.e., the first coin being a nickel and the second coin being a quarter) is the product of the individual probabilities:
[tex]\[
P(\text{First coin is a nickel and second coin is a quarter}) = P(\text{First coin is a nickel}) \times P(\text{Second coin is a quarter | First coin is a nickel}) = \frac{4}{13} \times \frac{1}{6} = \frac{4}{78} = \frac{2}{39} \approx 0.0513
\][/tex]
Therefore, the probability that the first coin is a nickel and the second coin is a quarter is [tex]\(\frac{2}{39}\)[/tex].
The correct answer is:
[tex]\[
\frac{2}{39}
\][/tex]
