a. In the lottery game, it seems that repetition of numbers getting picked is *not* allowed. There are
[tex]\dbinom{37}5=\dfrac{37!}{5!(37-5)!}=435,897[/tex]
ways of picking any 5 numbers between 1 and 37, so the probability of winning the jackpot is 1/435,897.
b. With repetition allowed, there are
[tex]10^5=100,000[/tex]
possible ways to draw any 5 numbers between 0 and 9, so the probability of winning the jackpot is 1/100,000.
c. In order for the game to be no-profit, its expected value should be 0. This basically means that the amount of money players pay to play the game should be balanced exactly by the winning amount. If [tex]X[/tex] represents the amount of money won by a player, then
[tex]X=\begin{cases}-1&\text{if the picked numbers do not match}\\49,999&\text{if the picked numbers do match}\end{cases}[/tex]
with probabilities
[tex]P(X=x)=\begin{cases}\dfrac{99,999}{100,000}&\text{for }X=-1\\\dfrac1{100,000}&\text{for }X=49,999\end{cases}[/tex]
The expected value is
[tex]E[X]=\displaystyle\sum_xx\,P(X=x)=-\dfrac{99,999}{100,000}+\dfrac{49,999}{100,000}=-\dfrac12[/tex]
which means the average player is expected to lose $0.50.
The game is no-profit if the jackpot is [tex]J[/tex] such that
[tex]E[X]=-\dfrac{99,999}{100,000}+\dfrac{J-1}{100,000}=0[/tex]
Solving for [tex]J[/tex] tells us the jackpot should be $100,000.
