Suppose you go down to the corner store to buy some lottery tickets.

Assumptions:

- Each ticket really consists of 8 chances for you to win money (that is each ticket really consists of you buying 8 bets). You have the possibility to win all 8 bets, but you also have the possibility to lose each time as well.

-Your allowed to purchase an infinite number of tickets, however you can only purchase one ticket at a time and each time you purchase another ticket, the probability of you winning decreases for that ticket.

- The first ticket you purchase is labelled $\displaystyle 1000$, your second ticket is labelled $\displaystyle 1001$, and similarly the $\displaystyle nth$ ticket that you purchase will be labelled $\displaystyle 1000 + n$

- The probability of you winning any one of the 8 individual bets on your ticket is equal to $\displaystyle \frac{1}{ \sqrt {\frac {1}{2}}{(1000+n)}}} $ ...............(note: the sqrt is suppose to cover $\displaystyle {\frac {1}{2}}{(1000+n)}$ and not just the $\displaystyle \frac{1}{2}$

..................................

Calculate the probability that someone will eventually purchase a ticket where all 8 bets on THAT ticket win? If the probability is $\displaystyle 100%$, what is the expected number of tickets which must be purchased for this event to occur?