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Someone told me something interesting today. They said they were going to take their bonus check from work, to the casino because they have "better odds" of paying off more debt then if they would just apply it to the principle balance of their outstanding credit card. Talking some more with the person they indicated they had about $20,000 in debt at about 20% interest.

Lets assume that their bonus check is $1000, they pay 500 a month towards their credit card, and they play a game with a 'better' chance of winning at the casino (craps, baccarat, etc) and have a 0.48 chance to win on the dollar.

We know their expected return is -0.04 per game, but since their owed interest and balance is so high; how would one go about calculating if their statement is true?

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  • $\begingroup$ *than${{{{}}}}$ $\endgroup$
    – Git Gud
    Commented May 5, 2015 at 13:07
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    $\begingroup$ Casinoes do their business on the fact that that statement is false. It's how they earn money. $\endgroup$
    – Arthur
    Commented May 5, 2015 at 13:10
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    $\begingroup$ Is there a math-related question here? $\endgroup$
    – hardmath
    Commented May 5, 2015 at 13:22
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    $\begingroup$ What might be true is that they have a positive probability of being able to completely pay off their debt, however having a positive probability and having it be likely are two very different things. As Arthur mentioned, the common phrase to hear is "The house always wins." By that we mean at the end of the day players have lost more money in total than they had won. Sure, there might be some lucky guy who made a fortune, but with how much everyone else lost it more than makes up for it. As such, you should expect to lose money and it is no substitute for proper investment. $\endgroup$
    – JMoravitz
    Commented May 5, 2015 at 13:22
  • $\begingroup$ Generally, the best money-making strategy is to put everything on a single red/black roulette spin. This has a negative expected value, of course. But if you consider a 47% chance of doubling the bonus (at which point one must walk away) and a 53% chance of the bonus evaporating, then... I guess it's a "good" idea. $\endgroup$
    – pjs36
    Commented May 5, 2015 at 13:25

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Clearly, this is not a wise decision in general, assuming constant utility per dollar. But let's explore it in brief, anyway.

We'll make some more assumptions. They put the entire grand on a single even-money bet, and win another grand with probability $0.48$, and lose their bet with probability $0.52$. That's it; then they walk home. (Obviously, this is not the way real people bet, but there's a limit to how much we can model. We just want to see if there's a non-linear effect one can take advantage of.)

On the debt side, they have debt at $20$ percent interest, and we'll assume zero inflation. (Now we're really fictional!) With those assumptions, then a debt of $D$ will get paid off in $n$ months according to

$$ \frac{500(1-r^n)}{1-r} = D $$ $$ 500(1-r^n) = D(1-r) $$ $$ 500r^n = 500-D(1-r) $$ $$ r^n = 1-\frac{D(1-r)}{500} $$

with $r = 60/61$. For $D = 20000$, this yields $n \doteq 64.513$; for $D = 19000$, $n \doteq 61.699$; for $D = 18000$, $n \doteq 59.009$. Paying down a thousand dollars of debt immediately saves your co-worker about $1357$ dollars; paying another thousand of winnings saves them only another $1345$ dollars. So there's a non-linearity, but it goes the wrong way. (Not surprising, really, since the further in debt you are, the more additional debt hurts you.)

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  • $\begingroup$ So with a probability of 0.52 to end up with 0 and a 0.48 chance to end up 4702 (1000 + 1000 + 1357 + 1345) would give us an expected value of 0.52 * 0 + 0.48 * 3702 of 2256.96? Versus if he applied it directly which would give us an expected value of 2357 (1000 + 1357). So we can expect a theoretical loss of of approximately 0.0424 (1 - 2256.96/2357)? $\endgroup$
    – aso118
    Commented May 5, 2015 at 19:40
  • $\begingroup$ No; when we assume the winnings are applied entirely to debt, the $1357$ and $1345$ are the positive impact of the $2000$. So if he gambles, he has a $48$ percent chance of ending up $1357+1345 = 2702$ better off, whereas if he doesn't gamble, he has a $100$ percent chance of ending up $1357$ better off. The theoretical loss would be $1357-2702\cdot 0.48 \doteq 60$. (I hasten to emphasize that this is "theoretical" only in a very loose sense.) $\endgroup$
    – Brian Tung
    Commented May 5, 2015 at 20:11

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