Answer to Question #295270 in Statistics and Probability for Hom

Question

The automatic opening device of a military cargo parachute has been designed to open

when the parachute is 202 m above the ground. Suppose opening altitude actually has a normal

distribution with mean value 202 m and standard deviation 31 m. Equipment damage will occur

if the parachute opens at an altitude of less than 102 m. What is the probability that there is

equipment damage to the payload of at least one of five independently dropped parachutes?

Answer 1

$\mu=202\\\sigma=31$

Let the random variable $X$ denote the event that equipment damage will occur. 

We first determine the probability that damage will occur given as,

$p(X\lt 102)=p({X-\mu\over\sigma}\lt {102-\mu\over \sigma})=p(Z\lt{102-202\over31})=p(Z\lt-3.23)=\phi(-3.23)=0.0006$The probability that there is no equipment damage is $1-0.0006=0.9994$  

Now the probability that there is equipment damage to the payload of at least one of five independently dropped parachutes is approximately $1 − (0.9994)^5=0.0029964$.