While writing her Ph D dissertation a doctoral student makes

While writing her Ph. D dissertation, a doctoral student makes a typo every 350 words. A typical page contains about 250 words. What is the probability that she will make at least 2 typos in 4 pages? Write the solution both in terms of a Binomial random variable as well as its approximation with a Poisson. Which of the two approaches is computationally easier and why? (You don\'t have to compute the harder of the two, just write out the ingredients of the solution.)

Solution

There is 1 typo every 350 words.

Thus, for a binomial distrbution:

n = 4*250 = 1000 words
p = 1/350 = 0.002857143
x >= 2

*********************

For a Poisson distribution:

There are, on average, 1*(250*4/350) = 2.857142857 typos in 4 pages.

Thus, we want P(x>=2).

*********************

The easier way is the Poisson way, so:

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    2.857142857      
          
x = our critical value of successes =    2      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   1   ) =    0.221525817
          
Thus, the probability of at least   2   successes is  
          
P(at least   2   ) =    0.778474183 [ANSWER]