To solve this problem, let's go through a detailed step-by-step solution.
1. Total Number of Coins:
First, calculate the total number of coins in your pocket. You have 4 nickels and 7 dimes.
[tex]\[
\text{Total Coins} = 4 (\text{nickels}) + 7 (\text{dimes}) = 11 \text{ coins}
\][/tex]
2. Number of Dimes:
You are given that there are 7 dimes in total.
3. Number of Canadian Coins:
There is 1 Canadian nickel and 1 Canadian dime, so the total number of Canadian coins is:
[tex]\[
\text{Canadian Coins} = 1 (\text{nickel}) + 1 (\text{dime}) = 2 \text{ coins}
\][/tex]
4. Calculate the Probability of Selecting a Dime:
The probability of selecting a dime is computed by dividing the number of dimes by the total number of coins.
[tex]\[
P(\text{Dime}) = \frac{\text{Number of Dimes}}{\text{Total Coins}} = \frac{7}{11}
\][/tex]
5. Calculate the Probability of Selecting a Canadian Coin:
The probability of selecting a Canadian coin is computed by dividing the number of Canadian coins by the total number of coins.
[tex]\[
P(\text{Canadian}) = \frac{\text{Number of Canadian Coins}}{\text{Total Coins}} = \frac{2}{11}
\][/tex]
6. Calculate the Probability of Selecting a Canadian Dime:
Since there is only 1 Canadian dime, the probability of selecting this specific coin is:
[tex]\[
P(\text{Canadian Dime}) = \frac{1}{11}
\][/tex]
7. Combine the Probabilities:
We are interested in the probability that the selected coin is either a dime or a Canadian coin. We use the principle of inclusion-exclusion to combine these probabilities:
[tex]\[
\text{Probability} = P(\text{Dime}) + P(\text{Canadian}) - P(\text{Canadian Dime})
\][/tex]
Substituting the values calculated:
[tex]\[
\text{Probability} = \frac{7}{11} + \frac{2}{11} - \frac{1}{11}
\][/tex]
Simplify the expression:
[tex]\[
\text{Probability} = \frac{7 + 2 - 1}{11} = \frac{8}{11}
\][/tex]
The final probability that you will either select a dime or a Canadian coin is:
[tex]\[
\boxed{\frac{8}{11}}
\][/tex]
So, the correct answer is B. [tex]\(\frac{8}{11}\)[/tex].
