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An anti aircraft gun can take a maximum of four shots at an enemy plane moving away from it. The probabilities of hitting the plane at first, second, third and fourth shot are 0.4, 0.3, 0.2 and 0.1 respectively. The probabality that the gun hits the plane then is
- a)0.6972
- b)0.6978
- c)0.6976
- d)0.6974
Correct answer is option 'C'. Can you explain this answer?
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An antiaircraft gun can take a maximum of four shots at an enemy plane...
Probability of hitting the plane on the first shot = 0.4
Probability of missing the plane on the first shot = 1 - 0.4 = 0.6
Probability of hitting the plane on the second shot = 0.3
Probability of missing the plane on the second shot = 1 - 0.3 = 0.7
Probability of hitting the plane on the third shot = 0.2
Probability of missing the plane on the third shot = 1 - 0.2 = 0.8
Probability of hitting the plane on the fourth shot = 0.1
Probability of missing the plane on the fourth shot = 1 - 0.1 = 0.9
In order to calculate the probability of hitting the plane, we need to consider all possible combinations of hitting and missing the plane on each shot. There are a total of 16 possible combinations (2^4 = 16).
Let's calculate the probability for each combination:
1. HHHH (all shots hit): 0.4 * 0.3 * 0.2 * 0.1 = 0.0024
2. HHHM (first three shots hit, last shot misses): 0.4 * 0.3 * 0.2 * 0.9 = 0.0216
3. HHMH (first two shots hit, third shot misses, fourth shot hits): 0.4 * 0.3 * 0.8 * 0.1 = 0.0096
4. HMHH (first two shots hit, third shot misses, fourth shot hits): 0.4 * 0.7 * 0.2 * 0.1 = 0.0056
5. MHMH (first shot hits, second shot misses, third shot hits, fourth shot misses): 0.6 * 0.3 * 0.8 * 0.1 = 0.0144
6. MMHH (first shot misses, second shot misses, third shot hits, fourth shot hits): 0.6 * 0.7 * 0.2 * 0.1 = 0.084
7. MHMH (first shot misses, second shot hits, third shot misses, fourth shot hits): 0.6 * 0.3 * 0.8 * 0.1 = 0.0144
8. MMMH (first three shots miss, last shot hits): 0.6 * 0.7 * 0.8 * 0.1 = 0.0336
9. HMHM (first two shots hit, third shot misses, fourth shot hits): 0.4 * 0.7 * 0.2 * 0.1 = 0.0056
10. HHMM (first two shots hit, third shot misses, fourth shot misses): 0.4 * 0.3 * 0.8 * 0.9 = 0.0864
11. HMHM (first shot hits, second shot misses, third shot hits, fourth shot misses): 0.6 * 0.3 * 0.2 * 0.9 = 0.0324
12. MMHM (first shot misses, second shot misses, third shot
Probability of missing the plane on the first shot = 1 - 0.4 = 0.6
Probability of hitting the plane on the second shot = 0.3
Probability of missing the plane on the second shot = 1 - 0.3 = 0.7
Probability of hitting the plane on the third shot = 0.2
Probability of missing the plane on the third shot = 1 - 0.2 = 0.8
Probability of hitting the plane on the fourth shot = 0.1
Probability of missing the plane on the fourth shot = 1 - 0.1 = 0.9
In order to calculate the probability of hitting the plane, we need to consider all possible combinations of hitting and missing the plane on each shot. There are a total of 16 possible combinations (2^4 = 16).
Let's calculate the probability for each combination:
1. HHHH (all shots hit): 0.4 * 0.3 * 0.2 * 0.1 = 0.0024
2. HHHM (first three shots hit, last shot misses): 0.4 * 0.3 * 0.2 * 0.9 = 0.0216
3. HHMH (first two shots hit, third shot misses, fourth shot hits): 0.4 * 0.3 * 0.8 * 0.1 = 0.0096
4. HMHH (first two shots hit, third shot misses, fourth shot hits): 0.4 * 0.7 * 0.2 * 0.1 = 0.0056
5. MHMH (first shot hits, second shot misses, third shot hits, fourth shot misses): 0.6 * 0.3 * 0.8 * 0.1 = 0.0144
6. MMHH (first shot misses, second shot misses, third shot hits, fourth shot hits): 0.6 * 0.7 * 0.2 * 0.1 = 0.084
7. MHMH (first shot misses, second shot hits, third shot misses, fourth shot hits): 0.6 * 0.3 * 0.8 * 0.1 = 0.0144
8. MMMH (first three shots miss, last shot hits): 0.6 * 0.7 * 0.8 * 0.1 = 0.0336
9. HMHM (first two shots hit, third shot misses, fourth shot hits): 0.4 * 0.7 * 0.2 * 0.1 = 0.0056
10. HHMM (first two shots hit, third shot misses, fourth shot misses): 0.4 * 0.3 * 0.8 * 0.9 = 0.0864
11. HMHM (first shot hits, second shot misses, third shot hits, fourth shot misses): 0.6 * 0.3 * 0.2 * 0.9 = 0.0324
12. MMHM (first shot misses, second shot misses, third shot
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An antiaircraft gun can take a maximum of four shots at an enemy plane moving away from it. The probabilities of hitting the plane at first, second, third and fourth shot are 0.4, 0.3, 0.2 and 0.1 respectively. The probabality that the gun hits the plane then isa)0.6972b)0.6978c)0.6976d)0.6974Correct answer is option 'C'. Can you explain this answer?
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