Can we buy more tickets to get more chances of winning the lottery?
The host can design a game with a probability of p and an award of R.
Assume that N players will come. The ticket for each is T. How much should T be?
There is a cost for each game, C.
NT - NpR - C > 0
Thus, the host can expect the income N(T-pR)-C.
T must be larger than C/N+pR, T > C/N+pR, otherwise the host loses money anyway.
For a player, one will be attracted if R > T and be very attracted to buy a ticket if R is much larger than T, R >> T.
When N is large and very large, the outcome will be close to the expected value, N(T-pR)-C. The more N, the closer the outcome. The host can obtain the expected outcome when N is large.
What is the expected outcome for the player?
Suppose that he buy n tickets.
He spends nT and expects to gain npR. The net gain is expected to be npR-nT or n(pR-T).
Note that, T is larger than pR, that n(pR-T) is negative. The net expected gain is negative for the player, n(pR-T) < 0. In other words, the outcome is expected to close to this negative value when the player buys more tickets.
For the host, he tries all efforts to attract more players, makes N large.
He knows that there must be some player who gets the award. However, he does not care who this lucky person is because his expected income is N(T-pR)-C.
The lucky one might be he, she, or YOU. Yes, it might be YOU.
If you think this way, you are trapped in "over-weighting of the rare event".
