A can hit a target 4 times out of 5 trial. B can hit 3 times of 4 trials and C can hit 2 times out of 3 trials. If all three hit the target sumultaneously, find the probability of hitting the target.

Explanation

To solve the problem step by step, we will calculate the probability of at least one of A, B, or C hitting the target.

### Step 1: Determine the probabilities of success for A, B, and C.

- **Probability of A hitting the target (P(A))**:
A hits 4 times out of 5 trials.
\[ P(A) = \frac{4}{5} \]

- **Probability of B hitting the target (P(B))**:
B hits 3 times out of 4 trials.
\[ P(B) = \frac{3}{4} \]

- **Probability of C hitting the target (P(C))**:
C hits 2 times out of 3 trials.
\[ P(C) = \frac{2}{3} \]

### Step 2: Calculate the probabilities of failure for A, B, and C.

- **Probability of A missing the target (P(A'))**:
A misses 1 time out of 5 trials.
\[ P(A') = 1 - P(A) = 1 - \frac{4}{5} = \frac{1}{5} \]

- **Probability of B missing the target (P(B'))**:
B misses 1 time out of 4 trials.
\[ P(B') = 1 - P(B) = 1 - \frac{3}{4} = \frac{1}{4} \]

- **Probability of C missing the target (P(C'))**:
C misses 1 time out of 3 trials.
\[ P(C') = 1 - P(C) = 1 - \frac{2}{3} = \frac{1}{3} \]

### Step 3: Calculate the probability of all three missing the target.

Since the events are independent (the success or failure of one does not affect the others), we can multiply the probabilities of failure:
\[ P(A' \cap B' \cap C') = P(A') \times P(B') \times P(C') \]
Substituting the values:
\[ P(A' \cap B' \cap C') = \frac{1}{5} \times \frac{1}{4} \times \frac{1}{3} = \frac{1}{60} \]

### Step 4: Calculate the probability of at least one hitting the target.

The probability of at least one hitting the target is given by:
\[ P(\text{at least one hits}) = 1 - P(A' \cap B' \cap C') \]
Substituting the value we found:
\[ P(\text{at least one hits}) = 1 - \frac{1}{60} = \frac{60}{60} - \frac{1}{60} = \frac{59}{60} \]

### Final Answer:
The probability of at least one of A, B, or C hitting the target is:
\[ \frac{59}{60} \]