Normal approximation to poisson got this question to do but I'm totally stuck. 
It goes as follows :
People call car breakdown service at an average rate of 20 per HR. A Poisson distribution may be assumed to model this situation. 
1) by using a suitable approximation find the probability that exactly 250 people call in 12 hrs. 
So I know that it follows a Poisson distribution with mean of 240. Ideally I think this is a normal approximation to the Poisson s distribution since mean>15 however the question asks for exactly 250 and as far as I know normal can't do exactly values. Also I don't see how to get a binomial from this since we don't have a value of n
 A: Exact Poisson: First adjust the rate to 240 to match 12 hrs. Then let $X$ be the number
of calls in 12 hrs, where $X \sim \mathsf{Pois}(240)$ and you seek
$P(X = 250) = 0.0205.$ (Computation in R, but computation using the Poisson
PDF, or PMF, isn't difficult on a calculator.)
dpois(250, 240)
[1] 0.02053754

Normal approximation: You have $\mu = E(X) = 240$ and $\sigma^2 = Var(X) = 240.$ So 
$\sigma = \sqrt{240} = 15.4919.$ So $$P(X = 240) \approx P(249.5 < Y < 250.5),$$ where $Y \sim \mathsf{Norm}(\mu = 240, \sigma=15.4919).$ Below is the computation of this normal
approximation using R. I suppose you are expected to get the answer by
standardizing and using normal tables. See how close you can get. [It may not be exact because a little rounding may be necessary to use printed normal tables.]
diff(pnorm(c(249.5, 250.5), 240, 15.4919))
[1] 0.02090658

Binomial approximation. Don't give up on using a binomial distribution. When $n$ is very large and $p$ is small, $\mathsf{Binom}(n, p)$ is well
approximated by $\mathsf{Pois}(\lambda = np).$ Let's try $n = 2400$ and $p = 1/10.$
dbinom(250, 2400, 1/10)
[1] 0.02120139

Or, maybe better, $n = 24,000$ and $p = 1/100.$
dbinom(250, 24000, 1/100)
[1] 0.02060194

Binomial probabilities can be a little messy to compute on a calculator because the factorials
in the binomial coefficient are so large.
The plot below shows the Poisson distribution (black bars, values between 230 and 260), the approximating normal density curve (blue), and the second binomial approximation (purple circles).

Note: In any case, it is useful to know relationships among binomial, Poisson, and normal distributions.
