Question

The mean life of a power station is 30 years and follows an exponential distribution.

1.What is the chance that the power station is inoperative? If it's maximum life is 50 years.

2. If six power stations are operated independently, what is the chance that at least 4 will still stand

after 40 years?

Answer 1

Let $X=$  life of a power station: $X\sim Exp(\lambda).$

$\mu=30=>\lambda=\dfrac{1}{\mu}=\dfrac{1}{30}$

1.

$$P(X\leq 50)=1-e^{-{1 \over 30}(50)}=1-e^{-{5 \over 3}}\approx0.8112$$

The chance that the power station is inoperative is 0.8112.

2.

$P(X\geq40)=1-P(X<40)=1-(1-e^{-{1 \over 30}(40)})=e^{-{4 \over 3}}\approx0.8112$

$$=e^{-{4 \over 3}}\approx0.2636$$

Let $Y=$ the number of lamps working after 40 years: $Y\sim Bin(n, p).$

$p=0.2636, n=6, q=1-p=1-0.2636=0.7364$

$$P(Y\geq 4)=P(Y=4)+P(Y=5)+P(Y=6)$$

$$=\dbinom{6}{4}(0.2636)^4(0.7364)^2+\dbinom{6}{5}(0.2636)^5(0.7364)^1$$

$$+\dbinom{6}{6}(0.2636)^6(0.7364)^0\approx0.045232$$

The chance that at least 4 from 6 power stations will still stand after 40 years is $4.5\%.$