g The average lifespan for a certain type of vehicle is 8 years and follows an exponential distribution. A lot contains 200 of these vehicles, brand new. What is the probability that 50 or more of them fail in the first 2 years

The probability is

Step-by-step explanation:

The average life span is  

The sample size is n =  200

Generally we are told that average lifespan for a certain type of vehicle follows an exponential distribution

i.e  

Gnerally the probability distribution function for exponential distribution is

Now is the  constant which is evaluated as

=>  

=>  

Generally the that the vehicles will fail in less than 2 years is  mathematically represented as

Generally the probability that the vehicle will fail in the first 2 years follows a binomial  distribution  with the mean evaluated as

=>      

=>      

The standard deviation is

Using normal approximation of binomial  distribution the  probability that 50 or more of them fail in the first 2 years is mathematically represented as

Applying continuity correction

Here

Generally  

=>

From the z-table the probability value  of  ( Z  <  1.0664   ) is  

So  

=>