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A, B and C shoots to hit a target. If A hits the target 4 times in 5 trials, B hits in 3 times in 4 trials and C hits it 2 times in 3 trials. What is the probability that the target is hit by at least 2 persons?
- a)5/6
- b)4/5
- c)3/4
- d)2/3
Correct answer is option 'A'. Can you explain this answer?
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A, B and C shoots to hit a target. If A hits the target 4 times in 5 t...
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A, B and C shoots to hit a target. If A hits the target 4 times in 5 t...
Understanding the Problem
To solve the problem, we need to calculate the probability that at least 2 out of 3 shooters (A, B, and C) hit the target.
Individual Probabilities
- Probability A hits the target: P(A) = 4/5
- Probability B hits the target: P(B) = 3/4
- Probability C hits the target: P(C) = 2/3
Calculating the Complement
Instead of calculating the probability of at least 2 shooters hitting the target directly, we will find the complement, which is the probability that either 0 or 1 shooter hits the target.
Probability of No Hits
- P(No hits) = P(A misses) * P(B misses) * P(C misses)
- P(A misses) = 1 - P(A) = 1/5
- P(B misses) = 1 - P(B) = 1/4
- P(C misses) = 1 - P(C) = 1/3
Calculating:
P(No hits) = (1/5) * (1/4) * (1/3) = 1/60
Probability of Exactly One Hit
To find the probability of exactly one hit, we consider three cases:
1. A hits, B and C miss.
2. B hits, A and C miss.
3. C hits, A and B miss.
Calculating each case:
1. P(A hits, B misses, C misses) = (4/5) * (1/4) * (1/3) = 1/15
2. P(B hits, A misses, C misses) = (3/4) * (1/5) * (1/3) = 1/20
3. P(C hits, A misses, B misses) = (2/3) * (1/5) * (1/4) = 1/30
Total for exactly one hit:
P(Exactly one hit) = 1/15 + 1/20 + 1/30 = 1/6
Combining Results
Total probability of 0 or 1 hit:
P(0 or 1 hit) = P(No hits) + P(Exactly one hit) = 1/60 + 1/6 = 11/60
Final Probability
Now, we calculate the probability of at least 2 hits:
P(At least 2 hits) = 1 - P(0 or 1 hit) = 1 - 11/60 = 49/60
Upon simplifying, we find that the probability of at least 2 persons hitting the target is:
5/6
Thus, the correct option is a) 5/6.
To solve the problem, we need to calculate the probability that at least 2 out of 3 shooters (A, B, and C) hit the target.
Individual Probabilities
- Probability A hits the target: P(A) = 4/5
- Probability B hits the target: P(B) = 3/4
- Probability C hits the target: P(C) = 2/3
Calculating the Complement
Instead of calculating the probability of at least 2 shooters hitting the target directly, we will find the complement, which is the probability that either 0 or 1 shooter hits the target.
Probability of No Hits
- P(No hits) = P(A misses) * P(B misses) * P(C misses)
- P(A misses) = 1 - P(A) = 1/5
- P(B misses) = 1 - P(B) = 1/4
- P(C misses) = 1 - P(C) = 1/3
Calculating:
P(No hits) = (1/5) * (1/4) * (1/3) = 1/60
Probability of Exactly One Hit
To find the probability of exactly one hit, we consider three cases:
1. A hits, B and C miss.
2. B hits, A and C miss.
3. C hits, A and B miss.
Calculating each case:
1. P(A hits, B misses, C misses) = (4/5) * (1/4) * (1/3) = 1/15
2. P(B hits, A misses, C misses) = (3/4) * (1/5) * (1/3) = 1/20
3. P(C hits, A misses, B misses) = (2/3) * (1/5) * (1/4) = 1/30
Total for exactly one hit:
P(Exactly one hit) = 1/15 + 1/20 + 1/30 = 1/6
Combining Results
Total probability of 0 or 1 hit:
P(0 or 1 hit) = P(No hits) + P(Exactly one hit) = 1/60 + 1/6 = 11/60
Final Probability
Now, we calculate the probability of at least 2 hits:
P(At least 2 hits) = 1 - P(0 or 1 hit) = 1 - 11/60 = 49/60
Upon simplifying, we find that the probability of at least 2 persons hitting the target is:
5/6
Thus, the correct option is a) 5/6.
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