Answer to Question #345925 in Statistics and Probability for bethelehem girma

Question

Suppose that lifetimes of light bulbs produced by a certain company are normal random

variables with mean 1000 hours and standard deviation 100 hours. Suppose that

lifetimes of light bulbs produced by a second company are normal random variables

with mean 900 hours and standard deviation 150 hours. A person buys one light bulb

manufactured by the first company and one by the second company. What is the

probability that at least one of them lasts 980 or more hours?

Answer 1

$$P(X_1<980)=P(Z_1<\dfrac{980-1000}{100})$$

$$=P(Z_1<-0.2)\approx0.42074$$

$$P(X_2<980)=P(Z_2<\dfrac{980-900}{150})$$

$$\approx P(Z_1<0.533333)\approx0.70310$$

$$P(at\ least\ one)=1-P(none)$$

$$=1-P(X_1<980)P(X_2<980)$$

$$\approx1-0.42074(0.70310)$$

$$\approx0.7042$$