If I buy one lottery ticket with odds of $14M$ to one if I buy another ticket with different numbers does this slash the odds to $7M$ to one? if so it follows that if I double my tickets again to $4$ Tickets it will half again to $3.5M$ to one?
So $8$ tickets will be $1.75M$ to one.
$16$ Tickets $\rightarrow.875M$ to one
$32 \rightarrow .4475$M to one
$64 \rightarrow 218750$ to one
$128 \rightarrow 109375$ to one
$256 \rightarrow 54587$ to one
Something seems wrong here, i am not looking for a exact answer just ballpark working out.
Best Answer
The odds are in this case given by $$\text{number of tickets you have}:{\text{number of tickets you don't have}}.$$
The important thing is that the denominator is not $\text{total number of tickets}$. So if you have $2^n$ tickets and the total number of tickets is $T,$ the odds of you winning will be $$2^n:T-2^n \iff 1:\frac{T}{2^n}-1,$$
which is different to your calculations. Let's take a look:
$n = 1$ gives odds of $ 1:\frac{T}{2}-1$
$n =2$ gives odds of $1:\frac{T}{4}-1,$
and so on, which grows much slower than your solution.
Edit: Here you can see a graph of your odds of winning as a function of the amount of $n$, the number of times you double your number of tickets. As you can see, it does actually grow exponentially, but it is off to an extremely slow start. At around $n=24$, you have bought all the tickets, so here the odds are $1:0,$ which gives the singularity shown.
So the bottom line is: Don't play the lottery! ;)