The probability that a certain beginner at golf gets a good shot if he uses the correct club is `1/3`, and the probability of a good shot with an incorrect club is `1/4`. In his bag are 5 different clubs, only one of which is correct for the shot is question. if he chooses a club at random and takes a stroke, the probability that he gets a good shot is

Options

A. `1/3`
B. `1/(12)`
C. `(4)/(15)`
D. `7/(12)`

Answer

To find the probability that the beginner at golf gets a good shot when he chooses a club at random, we can use the law of total probability.

### Step-by-Step Solution:

1. **Identify the probabilities:**
- Probability of choosing the correct club (C): \( P(C) = \frac{1}{5} \) (since there is 1 correct club out of 5)
- Probability of choosing an incorrect club (I): \( P(I) = \frac{4}{5} \) (since there are 4 incorrect clubs out of 5)
- Probability of a good shot with the correct club: \( P(G | C) = \frac{1}{3} \)
- Probability of a good shot with an incorrect club: \( P(G | I) = \frac{1}{4} \)

2. **Calculate the probability of getting a good shot:**
- Using the law of total probability:
\[ P(G) = P(G | C) \cdot P(C) + P(G | I) \cdot P(I) \]
- Substitute the values:
\[ P(G) = \left(\frac{1}{3} \cdot \frac{1}{5}\right) + \left(\frac{1}{4} \cdot \frac{4}{5}\right) \]

3. **Perform the calculations:**
- Calculate \( P(G | C) \cdot P(C) \):
\[ P(G | C) \cdot P(C) = \frac{1}{3} \cdot \frac{1}{5} = \frac{1}{15} \]
- Calculate \( P(G | I) \cdot P(I) \):
\[ P(G | I) \cdot P(I) = \frac{1}{4} \cdot \frac{4}{5} = \frac{4}{20} = \frac{1}{5} \]

4. **Combine the results:**
- Now add the two probabilities:
\[ P(G) = \frac{1}{15} + \frac{1}{5} \]
- Convert \( \frac{1}{5} \) to a fraction with a denominator of 15:
\[ \frac{1}{5} = \frac{3}{15} \]
- Therefore:
\[ P(G) = \frac{1}{15} + \frac{3}{15} = \frac{4}{15} \]

5. **Final Result:**
- The probability that he gets a good shot is \( \frac{4}{15} \).