can someone check my solutions to the first 2 parts?

i used binomial distribution for a) and poisson distribution for b) and they came out pretty close. so since the poisson distribution is an approximation of the binomial distribution, we would expect them to be very similar?

I don't know if this is the right approach, but anyway here goes:

a) If everybody is to be boarded there must be at most 90 people out of the 95 that shows up. On the other hand 89, 88, 87... people is also fine; so, to get the probability in question you need to sum up the probabilities that n people shows up for [tex]n = 0,\ldots,90[/tex] - like this:

(this tedious calculation can be done using the binomial distribution function available on most calculators)

b) The approximation is a Poisson-distribution with [tex]\lambda = np = 95 \cdot 0.95[/tex]. Again; you should use the Poisson distribution function (not just the density function with x = 90) since the number of passengers that shows up can be anything from 0 to 90.

c) The approximation is a [tex]N(np,np(1-p))[/tex]-distribution and similar to a) and b) you should use the distribution function to calculate the probability.

I don't how to "correct for continuity"...

e) I think the most effective way is just trial and error. Use the expression in a) with different values of n (instead of [tex]n = 95[/tex]) or use one of the approximations in b) or c).