An anti-aircraft gun can take a maximum of 4 shots at an enemy plane moving…
2021
An anti-aircraft gun can take a maximum of 4 shots at an enemy plane moving away from it. The probabilities of hitting the plane at the first, second, third and fourth shots are 0.6, 0.5, 0.4 and 0.2 respectively. What is the probability that the gun will hit the plane?
- A.
0.904
- B.
0.924
- C.
0.804
- D.
0.824
Attempted by 13 students.
Show answer & explanation
Correct answer: A
Key idea: compute the probability of at least one hit. It is simplest to use the complement: 1 minus the probability that all four shots miss.
Probability first shot misses = 1 - 0.6 = 0.4.
Probability second shot misses = 1 - 0.5 = 0.5.
Probability third shot misses = 1 - 0.4 = 0.6.
Probability fourth shot misses = 1 - 0.2 = 0.8.
Probability all four miss = 0.4 * 0.5 * 0.6 * 0.8 = 0.096.
Using the complement: probability of at least one hit = 1 - 0.096 = 0.904.
Verification by direct summation of mutually exclusive hit events:
Hit on first shot: 0.6
Miss first, hit second: 0.4 * 0.5 = 0.20
Miss first two, hit third: 0.4 * 0.5 * 0.4 = 0.08
Miss first three, hit fourth: 0.4 * 0.5 * 0.6 * 0.2 = 0.024
Sum = 0.6 + 0.20 + 0.08 + 0.024 = 0.904.
Answer: 0.904