A, B and C shoot to hit a target. If A hits the target 4 times in 5 trials, B hits it 3 times in 4 trials and C hits it 2 times in 3 trials. What is the probability that the target is hit by atleast 2 persons?

Options

A. `5/6`
B. `3/4`
C. `4/5`
D. `1/9`

Answer

To solve the problem, we need to calculate the probability that at least 2 out of the 3 shooters (A, B, and C) hit the target. We will first find the probabilities of each shooter hitting or missing the target, and then we will calculate the required probability using complementary counting.

### Step 1: Determine the probabilities of hitting and missing for each shooter.
- **Shooter A**: Hits the target 4 times in 5 trials.
- Probability of hitting (P(A)) = 4/5
- Probability of missing (P(A')) = 1 - P(A) = 1/5

- **Shooter B**: Hits the target 3 times in 4 trials.
- Probability of hitting (P(B)) = 3/4
- Probability of missing (P(B')) = 1 - P(B) = 1/4

- **Shooter C**: Hits the target 2 times in 3 trials.
- Probability of hitting (P(C)) = 2/3
- Probability of missing (P(C')) = 1 - P(C) = 1/3

### Step 2: Calculate the probability that at least 2 persons hit the target.
To find the probability that at least 2 persons hit the target, we can use the complement rule. We will first calculate the probability that fewer than 2 persons hit the target (i.e., either 0 or 1 person hits) and then subtract this from 1.

#### Case 1: Probability that no one hits the target.
- P(A') * P(B') * P(C') = (1/5) * (1/4) * (1/3) = 1/60

#### Case 2: Probability that exactly one person hits the target.
- **A hits, B and C miss**: P(A) * P(B') * P(C') = (4/5) * (1/4) * (1/3) = 4/60
- **B hits, A and C miss**: P(A') * P(B) * P(C') = (1/5) * (3/4) * (1/3) = 1/20 = 3/60
- **C hits, A and B miss**: P(A') * P(B') * P(C) = (1/5) * (1/4) * (2/3) = 1/30 = 2/60

Now, we sum the probabilities of the cases where exactly one person hits:
- Total probability for exactly one hit = (4/60) + (3/60) + (2/60) = 9/60

### Step 3: Calculate the total probability of fewer than 2 hits.
- Total probability for fewer than 2 hits = Probability of no hits + Probability of exactly one hit
- Total = (1/60) + (9/60) = 10/60 = 1/6

### Step 4: Calculate the probability that at least 2 persons hit the target.
- Probability(at least 2 hits) = 1 - Probability(fewer than 2 hits)
- Probability(at least 2 hits) = 1 - (1/6) = 5/6

### Final Answer:
The probability that at least 2 persons hit the target is **5/6**.

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