Last week, as you recall we introduced the idea of playing one-spot keno tickets, and the various rationales for doing so. This week we’ll take a closer look at playing a straight one-spot for $1, game after game. To gauge our chances, we must use a branching probability chart as below. The first column gives the game number that each branch stops on. For instance, if the length of run is 1, then that means that we hit our one-spot on the first game. If the length of run is 10, that implies that we lost our first nine tickets, but that we hit on the tenth game. The odds for one against gives us the odds against a run of any particular length, the return tells us what our net win or loss will be for each run length, and the return per branch column gives us our total expectation for each branch over an extended period of play.
There are three facts that stick out immediately on the chart. 1) The frequency of all the branches adds to 1, which is a check of our analysis, 2) Out of all the branches, only the first three provide a positive return, and 3) The total expected return for all the branches is zero. Thus if we play enough one-spots we are confident that we will go broke, if not sooner, then later. This is true because the expected return per branch converges on zero, and thus the sum is converged on as well.
Alas, this will be true of any amount we bet, whether it is $1, $10, or $100 per game, as long as we are wagering a fixed amount per game. This also is true no matter what our initial bankroll is, even a million bucks. The probability tree below will whittle it down to size. Of course there are betting systems called "Martigales" which vary the amount bet per game and we will cover them in the future.
Alas, our conclusion today must be that, in the long run, playing straight one-spots is a sure road to ruin!
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