Answer to Question #211179 in Statistics and Probability for salman

Question

Three missiles are fired at a target. If the probabilities of hitting the target are 0.3, 0.4, 0.6 respectively, and the missiles are fired independently, what is the probability that (i) all missiles hits the target (ii) exactly one hit the target (iii) no more than one hit the target?

Answer 1

By condition,

${p_1} = 0.3 \Rightarrow {q_1} = 1 - {p_1} = 0.7$

${p_2} = 0.4 \Rightarrow {q_2} = 1 - {p_2} = 0.6$

${p_3} = 0.6 \Rightarrow {q_3} = 1 - {p_3} = 0.4$

(i) Using the formula for multiplying probabilities, we get:

$p(3) = {p_1}{p_2}{p_3} = 0.3 \cdot 0.4 \cdot 0.6 = {\rm{0}}{\rm{.072}}$

Answer:0.072

(ii) By the forms of multiplication and addition of probabilities, we get:

$p(1) = {p_1}{q_2}{q_3} + {q_1}{p_2}{q_3} + {q_1}{q_2}{p_3} = 0.3 \cdot 0.6 \cdot 0.4 + 0.7 \cdot 0.4 \cdot 0.4 + 0.7 \cdot 0.6 \cdot 0.6 = {\rm{0}}{\rm{.072}} + {\rm{0}}{\rm{.112}} + {\rm{0}}{\rm{.252}} = {\rm{0}}{\rm{.436}}$

Answer: 0.436

(iii) Let's find the probability that 0 missiles will hit the target:

$p(0) = {q_1}{q_2}{q_3} = 0.7 \cdot 0.6 \cdot 0.4 = {\rm{0}}{\rm{.168}}$

Then the wanted probability is

$p(x \le 1) = p(0) + p(1) = 0.168 + 0.436 = 0.604$

Answer: 0.604