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There are 10 balls in a bucket with each ball marked 1 through 10 and are chosen at random from a bucket each day. The bucket is fair and all 10 balls are reset each day in the bucket. With a wager of 1:10 for guessing the ball number correctly, you win 10 dollars, or 9 dollars profit, for every $1 you bet. You can bet on only 1 of 10 numbers per day.

Now, let's say you also have the history for this particular game from inception for all 10 balls. A ball hasn't been selected from the bucket for over 50 consecutive days only 15 times in this game's entire history. Out of the 15 times this has happened, the ball was selected 14 of 15 times within the next 10 days. 1 of 15 was drawn on the 64th day. Currently, ball number 2 hasn't been selected from the bucket for 50 consecutive days. You are allowed to use the Martingale system or a similar system where you increase your wager each day you lose.

Does this mean winning at this point has a 93.3% (14/15) chance within the next 10 days? With the example above, would it be a good idea to make a bet on ball number 2 since it hasn't come out in 50 consecutive days?

wonderfulsomebody
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    I would first look at whether this is a statistically significant outcome given a null hypothesis of a true uniformly random draw, independent on each day. "The game's entire history" is not given, and that could hint toward a ball being more OR less likely when not recently drawn, depending. 14/15 is high (expected proportion around 79%), but doesn't seem extremely high. – aschepler Oct 04 '22 at 23:46
  • Each draw is independent so the odds are the same every day. The history doesn't matter. – John Douma Oct 04 '22 at 23:46
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    @JohnDouma That wasn't stated, although "chosen at random" could be misleading. – aschepler Oct 04 '22 at 23:47
  • @aschepler This is the gambler's ruin. – John Douma Oct 04 '22 at 23:48
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    @JohnDouma Not if the actual process of putting balls back in and taking them out makes the draws significantly correlated (positive or negative), rather than independent. – aschepler Oct 04 '22 at 23:49
  • @aschepler "There are 10 balls in a bucket with each ball marked 1 through 10 and are chosen at random from a bucket each day." Maybe we should make sure wonderfulsomebody understands the gambler's ruin before indulging folk lore about gambling. – John Douma Oct 04 '22 at 23:53
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    Yes, I suggest editing the question to clarify whether "at random" means "ideal" randomness (uniform and independent) or potentially "imperfect" randomness. – aschepler Oct 04 '22 at 23:55

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No. If "the bucket is fair", then all the information about the game's history and the recent draws is entirely irrelevant. The probability ball two will be drawn next is $1/10$, because that's what "the bucket is fair" means.

The expected profit from any bet, no matter what ball you bet on, is zero:

$$ \frac{1}{10}(+9) + \frac{9}{10}(-1) = 0 $$

(However, some patterns of persistent gambling will eventually lead to losses even with an entirely "fair" game like this: See the Gambler's Ruin.)

aschepler
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  • Two questions:

    1. Does this mean the game's history is just a coincidence that the balls were selected 14 out of 15 times between the 50th and 60th day? It's truly insignificant because the game is fair?

    2. What if the expected profit for any bet was much higher, such as 1:40 where you get 40 dollars for every 1 dollar bet - would it still be Gambler's Ruin?

     – wonderfulsomebody Oct 06 '22 at 16:59