draw lotteries

Counting combinations

Order does not matter in a draw, so the number of distinct tickets is a combination rather than a permutation. Choosing six numbers from forty-nine gives C(49,6), which works out to 13,983,816 — a figure worth deriving once, because the same method gives every other tier and every other game format.

Lower tiers are computed the same way, by counting how many tickets match a given number of drawn balls and a given number of undrawn ones. The arithmetic is elementary; the results are not intuitive, which is why they are worth writing out.

total lines            C(49,6) = 13,983,816

match 6   C(6,6) x C(43,0) =        1  ->  1 in 13,983,816
match 5   C(6,5) x C(43,1) =      258  ->  1 in     54,201
match 4   C(6,4) x C(43,2) =   13,545  ->  1 in      1,032
match 3   C(6,3) x C(43,3) =  246,820  ->  1 in         57

check: the tiers plus all non-winning lines sum to 13,983,816

Expected value and the prize fund

A lottery's expected value is set by the share of ticket revenue returned as prizes. If half the money staked is paid out, the expected value of a ticket is half its price, and the house edge — using the same definition as everywhere else on this site — is fifty percent. That is an order of magnitude larger than a table game, and it is entirely public, because the prize fund share is normally fixed by the licence.

The reason the figure is tolerated is that a lottery is not sold as a wager. Its stake is small, its frequency is low, and a large fraction of the money is typically directed to public or charitable purposes by statute. Those are policy facts, not mathematical ones, but they explain why the same arithmetic is regulated so differently across game types.

Rollovers

When a jackpot is not won, the amount rolls into the next draw. That added money was staked in an earlier draw, so the following draw returns more than its own ticket revenue and its expected value rises. In a large enough rollover the expected value of a ticket can in principle approach or exceed its price.

It rarely stays there. High jackpots sell more tickets, more tickets mean more chance of a shared jackpot, and sharing divides the prize that made the ticket attractive. The correction is automatic and is a neat example of a market absorbing an edge.

Pari-mutuel pools

Two prize structures exist. Under fixed odds a prize is a stated amount, and the operator carries the risk of an unusual number of winners. Under a pari-mutuel pool the prize is a share of a fund, divided among however many winners there are, and the operator carries no risk at all because the payout is defined as a fraction of the money taken.

This is why pooled lotteries and totalisator betting can publish their deductions openly: with the margin taken off the top before the pool is divided, the return is determined by arithmetic rather than by pricing judgement.