Question

question : Suppose that three missiles are available to fire at three targets, called Target 1, Target 2, and Target 3.Each time a missile is fired at Target 1 it will hit it with probability 0.9. Each time a missile is fired at Target 2 it will hit it with probability 0.8. Each time a missile is fired at Target 3 it will hit it with probability 0.7. Missiles will be fired, one at a time, at Target 1 until it is hit. If Target 1 is hit, any remaining missiles will be fired at Target 2 until it is hit, and if Target 2 is hit before the last of the three missiles is fired, the last missile will be fired at Target 3. Letting H1 be the event that the first missile fired hits a target, and H2 be the event that the second missile fired hits a target, explain why H1 and H2 are not independent events.

Answer 1

Question: Suppose that three missiles are available to fire at three targets, called Target 1, Target 2, and Target 3. Each time a missile is fired at Target 1 it will hit it with probability 0.9. Each time a missile is fired at Target 2 it will hit it with probability 0.8. Each time a missile is fired at Target 3 it will hit it with probability 0.7. Missiles will be fired, one at a time, at Target 1 until it is hit. If Target 1 is hit, any remaining missiles will be fired at Target 2 until it is hit, and if Target 2 is hit before the last of the three missiles is fired, the last missile will be fired at Target 3. Letting H1 be the event that the first missile fired hits a target, and H2 be the event that the second missile fired hits a target, explain why H1 and H2 are not independent events.

Answer: Two events $A$ and $B$ are independent if and only if their joint probability equals the product of their probabilities: $P(A \cap B) = P(A)P(B)$. So we should prove that $P(H_1 \cap H_2) \neq P(H_1)P(H_2)$.

Let $T(i, j)$ be the event that a missile number $j$, fired at the Target $i$, hits it, $i = 1, 2, 3, j = 1, 2, 3$. $T(i, j)$ are all independent events. $P(T(1,j)) = 0.9$, $P(T(2,j)) = 0.8$, $P(T(3,j)) = 0.7$, $j = 1, 2, 3$.

a) $H_{1} \cap H_{2}$ is the event that the first missile hits Target 1, and then second missile hits Target 2. $P(H_{1} \cap H_{2}) = P(T(1,1) \cap T(2,2)) = P(T(1,1))P(T(2,2)) = 0.9 * 0.8 = 0.72$.

b) $H_{1}$ is the event that the first missile hits a Target 1. $P(H_{1}) = P(T(1,1)) = 0.9$.

c) $H_{2}$ is the event that the second missile hits a target. There two possible cases:

1) First missile hits a Target 1 and then second hits Target 2 - $T(1,1)\cap T(2,2)$.

2) First missile misses a Target 1 and then second hits Target 1 - $\overline{T(1,1)}\cap T(1,2)$.

$$H_{2} = \left(T(1,1)\cap T(2,2)\right) \cup (\overline{T(1,1)}\cap T(1,2)), \text{ and } \left(T(1,1)\cap T(2,2)\right) \cap (\overline{T(1,1)}\cap T(1,2)) = \emptyset.$$

Therefore, $P(H_{2}) = P(T(1,1)\cap T(2,2)) + P(\overline{T(1,1)}\cap T(1,2)) = P(T(1,1))P(T(2,2)) + (1 - P(T(1,1)))P(T(1,2)) = 0.9 * 0.8 + (1 - 0.9) * 0.9 = 0.81$.

Obtained, that $P(H_{1}) P(H_{2}) = 0.9 * 0.81 = 0.729$, which is not equal to $P(H_{1} \cap H_{2}) = 0.72$. Therefore, $H_{1}$ and $H_{2}$ are not independent events.