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 probabilitiy that he gets a good shot is

Options

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

Answer

To solve the problem, we need to calculate the probability that the beginner at golf gets a good shot when he randomly selects a club from his bag. We will break this down into steps.

### Step 1: Identify the probabilities
- Probability of a good shot with the correct club (P(Good | Correct)) = 1/3
- Probability of a good shot with an incorrect club (P(Good | Incorrect)) = 1/4

### Step 2: Determine the number of clubs
- Total clubs = 5
- Correct clubs = 1
- Incorrect clubs = 4

### Step 3: Calculate the probabilities of selecting each type of club
- Probability of selecting the correct club (P(Correct)) = Number of correct clubs / Total clubs = 1/5
- Probability of selecting an incorrect club (P(Incorrect)) = Number of incorrect clubs / Total clubs = 4/5

### Step 4: Use the Law of Total Probability
We can find the total probability of getting a good shot (P(Good)) by considering both cases (selecting a correct club and selecting an incorrect club):

\[ P(Good) = P(Good | Correct) \cdot P(Correct) + P(Good | Incorrect) \cdot P(Incorrect) \]

### Step 5: Substitute the values
Substituting the known values into the equation:

\[ P(Good) = \left(\frac{1}{3} \cdot \frac{1}{5}\right) + \left(\frac{1}{4} \cdot \frac{4}{5}\right) \]

### Step 6: Calculate each term
- For the correct club:
\[ P(Good | Correct) \cdot P(Correct) = \frac{1}{3} \cdot \frac{1}{5} = \frac{1}{15} \]

- For the incorrect club:
\[ P(Good | Incorrect) \cdot P(Incorrect) = \frac{1}{4} \cdot \frac{4}{5} = \frac{4}{20} = \frac{1}{5} \]

### Step 7: Combine the results
Now, we add the two probabilities together:

\[ P(Good) = \frac{1}{15} + \frac{1}{5} \]

To add these fractions, we need a common denominator. The least common multiple of 15 and 5 is 15.

Convert \(\frac{1}{5}\) to a fraction with a denominator of 15:

\[ \frac{1}{5} = \frac{3}{15} \]

Now we can add:

\[ P(Good) = \frac{1}{15} + \frac{3}{15} = \frac{4}{15} \]

### Final Answer
The probability that the beginner at golf gets a good shot is \(\frac{4}{15}\).

---