Using Poisson random variables to calculate this probability

[Karl Porter]

Homework Statement

A certain theory supposed that mistakes in cell division occur according to a Poisson process with rate 2.5 per year, an individual dies when 196 mistakes have occured.

Relevant Equations

c) the probability that an individual reaches age 90
d)find the probability that an individual died before age 67.2

I calculated the mean which is 78.4
And the Standard deviation is 5.6
I thought the answer would be (90^(-78.4)/78.4!)*e^-90
But looking back having a decimal factorial doesn't make sense

I have the numerical answers for c)= 0.019226
and d)=0.022750
but I my solution was wrong.

Any help on how this question is actually solved would be great

 

[andrewkirk]

The clue that your answer is wrong lies in the fact that you haven't used the given number of 196, even though the answer must depend on that - the greater the number of mistakes needed to kill, the longer the expected life.
What is the probability of having fewer than 196 events in a time period of length 90, given a Poisson frequency of 2.5?