What is the probability that out of 245 booked reservations that exactly 224 people show up for the flight? What is the probability that no more than 224 people show up? How are these two probabilities different?

Jul 21, 2016

$P \left(x = 224\right)$ is exactly
$P \left(x \le 224\right)$ is at most

$p \left(x = 224\right)$ close to 0
$p \left(x \le 224\right)$ close to 1

Explanation:

I suppose given no additional information the probability of a person showing up is 50/50 although I doubt that the probability of missing your flight is that high. In any case we need to find

$P \left(x = 224\right)$

for this a reasonable distribution is the binomial and we are interested in 224 success out of 245 thus

p(x=224) = (245!)/((245-224)!224!) (1/2)^224(1/2)^(245-224)

p(x=224) = (245!)/((21)!224!)(1/2)^(245) ~~2*10^(-44)

this is if we are to assume exactly.

Now at at most is equivalent to

$P \left(x \le 224\right)$

in this case we need to consider
$p \left(x = 224\right) + p \left(x = 223\right) + \ldots$ all the way down to $p \left(x = 1\right)$. So first question asks what is the probability of exactly while the other considers the entire range. Now we can use the Cumulative distribution thus

P(x<=224) = sum_(i=0)^224 (245!)/((245-i)!i!) (1/2)^i(1/2)^(245-i)

close to 1