People make risky choices involving probabilities of reward and loss.  Probability scares the pants off most people.   I'm not talking about flight insurance or Hurricane Andrew.  Rather, it's the math, the mind-numbing contingencies, permutations, combinations, binomial expansions, Type I vs. Type II errors, and . . . snore, snore.  But the approximate $66 million stake in a big drawing (e.g. 22 May 1999) is an opener to talk math.

Can picking certain numbers improve the return-to-risk on Florida's lotto?  If you're not a player, the game is to pick six numbers that match six numbered balls drawn supposedly randomly in a studio somewhere.  The numbers can be from 1 to 49, without replacement, that is, there's no point in picking the same number twice, because only one ball has that number.

Some numbers are drawn more frequently

According to the number 41 was drawn twelve times between 04 Jan. 1997 and 20 Dec. 1997, but the number 28 had the privilege only once.  Many people picked the 41 as part of their six numbers, figuring that if it works, don't fix it.  But other people picked the 28, figuring that its time had come.  If you believe that the future follows the past, you might also have picked the 47 (drawn 11 times) and 45 (a 10-times draw).  On the other hand, if you believe that every number has its day, you might have picked the 28 along with 22, 33, and 38, only two-time draws and, the idea goes, at the head of the line for last Saturday.  (A footnote:  33 was the number drawn least frequently since the lottery first started in 1988.  It was also one of the lucky six for the sixty mill on 27 Dec. 1997).

If the 41 was in the jackpot twelve times, and the 28 only once, it sure looks like there's a pattern.  It's just, what's the pattern?   Should you pick the 41 and similar past "winning" numbers or the 28 and other past losers?  (About half the time there is no 6-number payout, that is, nobody correctly chooses six numbers, and the pot grows for the next week.)

What does probability say?

The graph above and the table below show the distribution of 1997's hot and cold numbers (through 20 Dec. 1997), compared with an estimate from probability theory (see footnotes below on how this was computed).  The estimate was based on assuming the balls are all equally weighted, etc., etc., and have an equal chance of getting drawn.  Although twelve seems to be a large number of recurrences for one number, it's not unusual for one number to be pulled 12 out of 51 draws.  The table below estimates that on average 3.5 numbers should recur at least ten times, among 51 draws.  Upon observation of 1997 results, exactly three numbers were drawn ten or more times.

We must accept the hypothesis that in the Florida Lottery, a number's past performance is no prediction of its future performance.  By deduction, it doesn't matter what numbers you pick. 

Said another way, evaluation of 1997 performances fails to disprove the hypotheses of randomness and independence.  I know that sounds contorted, but science never really proves anything, it can only disprove things or fail to disprove things.

I won't tread on areas of faith.  If you know what numbers will win, bet them.  Scientific knowledge must always be disprovable, besides, if you show me $58 million, I too will believe in lucky numbers.

Footnotes on Calculating Florida Lotto probabilities with Microsoft Excel

1. On a single drawing, the frequency of a number being drawn is 12.2%, that is 6 chances in 49 that a particular ball will be in a group.  Assuming the balls have no affinity for one another, then 12.2% is the chance that a ball will be in the drawing.

2. The chance that a particular number ball is picked multiple times among weekly drawings is determined by the binomial distribution.   So you don't have to expand the binomial pyramid, Microsoft Excel has a function BINOMDIS(number_s, trials, probability, cumulative).  For the first 51 drawings of the Florida Lotto in 1997, you enter in the function window f_x=BINOMDIS(A1, 51, 0.122, FALSE), where A1 is the name of the cell whence you are taking a number from 0 to 51, indicating the number of times a number might be expected to win.  (FALSE is added to indicate that the results are not cumulative, but specific.) 

3. As a specific example, open Microsoft Excel to a New Document, and write down the A column the numbers from 0 to 51. Click to select the B1 cell, and then place the cursor in the f_x formula window above the spreadsheet.  Write in the window the formula above and then copy and paste it down all the cells of the B column.  Each time you do this, the formula knows that it's being copied to the B1 cell, B2 cell, B3, B4, etc. down to B52 so it has to look up a value (0, 1, 2, 3 . . . 51) from the corresponding A1, A2, A3, A4 down to A52 cells.  If you want to enhance this to base it on actual draws (not raw probability), you just multiply it by the total number of draws, 51, entering instead the formula:   f_x=BINOMDIS(A1, 51, 0.122, FALSE)*51   That's it!

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