Question

Laura and Philip each fire one shot at a target. Laura has probability 0.45 of hitting the target, and Philip has probability 0.33. The shots are independent

a. Find the probability that the target is hit.
b.Find the probability that the target is hit by exactly oneshot.
c.Given that the target was hit by exactly one shot, find theprobabilty that Laura hit the target.

Answer 1

Answer with Step-by-step explanation:

Let A=Laura

B=Philip

Laura has probability of hitting the target,P(L)=0.45

Philip has probability of hitting the target,P(B)=0.33

The shots are independent

When A and B are independent then

$P(A)\cdot P(B)=P(A\cap B)$

a.The probability that the target is hit

=$P(A\cup B)$

$P(A\cup B)=P(A)+P(B)-P(A\cap B)=P(A)+P(B)-P(A)\times P(B)$Substitute the values then we get

$P(A\cup B)$=$0.45+0.33-0.45\times 0.33=0.632$

Hence, the probability that the target is hit=0.632

b.The probability that the target is hit by exactly one shot=

$P(A\cap B')+P(A'\cap B)=P(A)\cdot P(B')+P(A')\cdot P(B)$

When A and B are independent

Then,$P(A\cap B')=P(A)\times P(B')$

The probability that the target is hit by exactly one shot=$0.45\times (1-0.33)+(1-0.45)\times 0.33$

By using P(A')=1-P(A)

The probability that the target is hit by exactly one shot=$0.3015+0.1815=0.483$

c.Given that the target was hit by exactly one shot,the probability that Laura hit the target=$\frac{P(A\cap B')}{P(A\cap B')+P(A'\cap B)}$

Given that the target was hit by exactly one shot,the probability that Laura hit the target=$\frac{0.45\times (1-0.33)}{0.483}=0.62$