The question asks about a contest that is run, with 5 million contestants, each of whom have a independent probability $p$ of winning. You must show that for any value of $p$ you choose there is a more than 40% chance that there will be either no winners or more than one winner.
(This is from Mathematical Methods for Physics and Engineering,3rd ed, Riley Hobson Bence, question 30.24)
When I attempted this question I calculated the probability of having one winner, and found it's critical points:
$P(X=1)=p*(1-p)^{4999999}$
$dP/dp=(1-p)^{4999998}(1+4999998p)$
$p=1/{4999998}$
And when I used this value I found:
$P(X=1)=7.3576*10^{-7}$
This agrees with the probability of having not exactly one winner is greater than 40% but the question seems rather misleading if the probability is actually 99% not and not closer to 40%.
Any hints or suggestions would be much appreciated!
You forgot to add the ${5000000\choose 1}=5000000$, so it should actually be $$P(X=1)=5000000p(1-p)^{4999999}$$
Also, $(1-p)^{4999998}(1+4999998p)$ should be $(1-p)^{4999998}(1-5000000p)$ (and then multiplied by $5000000$).