Question

The lifetime of a certain type of batteries follows an exponential distribution with the mean of 12 hours.

a) What is the probability that a battery will last more than 14 hours? (Answer: 0.3114)

b) Once a battery is depleted, it is replaced with a new battery of the same type. Assumingindependence between lifetimes of batteries, what is the probability that exactly 2 batteries will be depleted within 20 hours? (Answer: 0.2623)

c) What is the probability that it takes less than 30 hours until the fourth battery is needed? (Answer: 0.4562)

Answer 1

    Let X be the lifetime of the batteries                      
   X follows Exponential distribution with mean = 12 hours                      
   Hence, λ = 1/12 batteries fail per hour                      
                          
a)   To find P(a battery will last more than 14 hours)                      
   that is to find P(X > 14)                      
   We use Excel function EXPON.DIST to find the probability                      
   P(X > 14) = 1 - P(X < 14)                      
       = 1 - EXPON.DIST(14, 1/12, TRUE)                  
       = 1 - 0.6886                  
       = 0.3114                  
   P(a battery will last more than 14 hours) = 0.3114                      
                          
b)   To find P(exactly 2 batteries will be depleted within 20 hours)                      
   Let Y be the number of batteries depleting in 20 hours                      
   We have 1/12 batteries depleted per hour                      
   Thus mean number of batteries depleting in 20 hours = 20/12                      
   Y ~ Poisson distribution with λ = 20/12 batteries depleting in 20 hours                      
   P(exactly 2 batteries will be depleted within 20 hours)                      
   that is to find P(Y = 2)                      
   We use Excel function POISSON.DIST to find the probability                      
   P(Y = 2) = POISSON.DIST(2, 20/12, FALSE)                      
                   = 0.2623                      
   P(exactly 2 batteries will be depleted within 20 hours) = 0.2623                      
                          
c)   We have 1/12 batteries depleted per hour                      
   Thus 4 batteries will be depleted in 4*12 = 48 hours                      
   Let Y be the time for depletion of 4 batteries                      
   Y follows Exponential distribution with λ = 1/48                      
   To find P(Y < 30)                      
   We use Excel function EXPON.DIST to find the probability                      
   P(Y < 30) = EXPON.DIST(30, 1/48, TRUE)                      
       = 0.4647                  
   P(it takes less than 30 hours until the fourth battery is needed) = 0.4647