Question

An anti-aircraft gun can take a minimum of four shots at an enemy plane moving away from it. The probability of hitting the plane at the first, second, third and fourth shot are 0.4, 0.3, 0.2 and 0.1 respectively. What is the probability that the gun hits the plane?

Solution

Let E₁ , E₂ , E₃ , E₄ be the events of hitting the plane by 1st, 2nd, 3rd, 4th shot respectively. Then

straight P left parenthesis straight E subscript 1 right parenthesis space equals space 0.4 comma space space space straight P left parenthesis straight E subscript 2 right parenthesis equals space 0.3 comma space space space straight P left parenthesis straight E subscript 3 right parenthesis space equals space 0.2 comma space space space straight P left parenthesis straight E subscript 4 right parenthesis space equals space 0.1

therefore space space space space straight P left parenthesis straight E with bar on top subscript 1 right parenthesis space equals space 0.6 comma space space straight P left parenthesis straight E with bar on top subscript 2 right parenthesis space equals space 0.7 comma space space straight P left parenthesis straight E with bar on top subscript 3 right parenthesis space equals space 0.8 comma space space straight P left parenthesis straight E with bar on top subscript 4 right parenthesis space equals space 0.9

Probabililty that no shot hit the plane

equals space straight P left parenthesis straight E with bar on top subscript 1 space space straight E with bar on top subscript 2 space straight E with bar on top subscript 3 space stack straight E subscript 4 with bar on top right parenthesis equals space straight P left parenthesis straight E with bar on top subscript 1 right parenthesis thin space straight P left parenthesis straight E with bar on top subscript 2 right parenthesis space space straight P left parenthesis straight E with bar on top subscript 3 right parenthesis thin space straight P left parenthesis straight E with bar on top subscript 4 right parenthesis space equals space.6 space cross times space.7 space cross times space.8 space cross times space.9 space equals space.3024

therefore

probability that at least one shot hit the plane.

equals space 1 minus straight P left parenthesis straight E with bar on top subscript 1 space space straight E with bar on top subscript 2 space straight E with bar on top subscript 3 space straight E with bar on top subscript 4 right parenthesis equals space 1 minus.3024 space equals space.6976

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