Draw games and the 6/49 matrix

A draw game asks the player to choose a set of numbers from a fixed pool, and its mathematics is pure combinatorics. The number of distinct unordered selections of k items from a pool of n is the binomial coefficient, n factorial divided by the product of k factorial and (n minus k) factorial. Order does not matter in a draw game, which is why the division by k factorial appears: the same six numbers drawn in any sequence are the same ticket.

For a six-from-forty-nine matrix the calculation runs 49 x 48 x 47 x 46 x 45 x 44 divided by 720, which gives 13,983,816 distinct combinations. Every one of them is equally likely, so the chance of matching all six on a single selection is one in 13,983,816. This figure is not an estimate or a published claim; it is a consequence of the matrix, and anyone can reproduce it in a minute.

Changing the pool or the selection size moves the top-prize probability dramatically, because factorials grow very fast. Extending a six-number game from a pool of forty-nine to a pool of fifty-nine raises the combination count to well over forty million, roughly tripling the difficulty. Adding a separate bonus pool multiplies the count again by the size of that pool. Designers use this lever to control how often the top tier is won, which in turn controls how often the prize rolls over.

Rollover is the point of the exercise. A game whose top prize is won most weeks never accumulates a headline figure; a game whose top prize is rarely won accumulates one that draws far more participation than the underlying probability would suggest. The matrix is therefore chosen to produce a target frequency of rollovers rather than a target probability, and periodic matrix changes in long-running games are almost always adjustments to that frequency.

Lower tiers are counted the same way, by choosing which of the drawn numbers are matched and which of the undrawn numbers fill the remaining slots. For five matches from six in a six-from-forty-nine game, there are six ways to choose which five of the six drawn numbers appear and forty-three ways to choose the sixth from the undrawn pool, giving 258 combinations out of 13,983,816, or about one in 54,200. The same method gives every other tier.

Summing the tiers reveals the overall chance of winning anything at all, which in most published matrices is dominated entirely by the smallest tier. This is the draw-game version of the paytable observation made earlier on this board: the frequent, small outcomes carry nearly all of the probability mass while the headline tier carries nearly all of the attention.

Draw games generally distribute a fixed proportion of ticket sales among winners rather than paying a fixed amount per winning ticket. Under this pari-mutuel arrangement the operator's margin is set the moment the distribution proportion is chosen, and it does not depend on the outcome at all. If a tier is shared by many winners each receives less; if by one, that one receives the whole tier. The pool structure transfers all outcome risk to the participants.

One consequence is worth stating because it is genuinely counter-intuitive. Since the pool is divided among those holding the winning selection, the expected value of a particular combination depends on how many other people chose it. Selections built from dates cluster in the low numbers, and patterned selections cluster on the physical geometry of a ticket slip. This changes the expected share of a prize, not the probability of winning it, and it never makes the expected value positive.