# De Econometrist

De Econometrist neemt een statistische kijk op de wereld.

Statistiek Wiskunde

# Guaranteed profit gambling

With over 1.6 billion people gambling at some point during a given year and over 4.2 billion people having gambled at some point in their lives, it is safe to say that gambling is booming. The risk versus reward element gives players a small rush of excitement and this is also the reason why gambling is so addictive for a lot of people. Lots of social events also include gambling, for example a poker evening. Long story short: there is a big chance that you will be gambling at some point in your life. But if we gamble, we also want to win, right? So why not use our mathematical and probability skills to minimize the probability of losing? In this article, I will present to you the so-called martingale betting technique, which is commonly referred to as the ‘never-lose’ or ‘infinite wealth’ betting strategy. How does this technique work and is it really that safe? Let’s find out!

### What is martingale betting?

Martingale betting is focussed on 50/50 betting, that is, there is a 0.5 probability of winning and a 0.5 probability of losing. Thus, examples of events where we can apply martingale betting are coin flips or betting on either red or black coming up on a two colored die. The technique was invented in the 18th century in France. The strategy had the gambler double the bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake.The best way to show how the technique works is by giving a mathematical example. Suppose we’re betting ‘M’ dollars (M > 0) on a coin landing heads, where, if it lands head we get our bet doubled back. Obviously, a coin lands either heads or tails with a 50/50 probability, so we can apply the martingale technique. Suppose we lose the first bet, which implies that we lost ‘M’ dollars. We now double the amount of our previous bet, thus we bet ‘2M’ dollars. Suppose we lose again, which implies that we lost ‘2M + M = 3M’ dollars. We double the amount of our bet again, so we bet ‘2M*2=4M’ dollars. Since the probability of losing three times in a row is quite small (0.5*0.5*0.5 = 0.125) let us assume we won the bet which yields us ‘8M’ dollars. In total, we’ve now spent ‘M + 2M + 4M = 7M’ dollars and we won ‘8M’ dollars. Clearly, 8M > 7M for M > 0 and we made a profit of M dollars. The martingale technique was successful!

### Martingale technique where the odds are against you

Let me show you that the probability that you will lose all your money is even bigger if you apply the martingale betting technique on a bet where the probability of losing is even a tiny bit larger than the probability of winning. Let q be the probability of losing (q > 0.5), let B be the amount of the initial bet (B > 0) and let n be a finite number of bets the gambler can afford to lose. Clearly, the probability of losing all n bets is (i.i.d., multiplication rule). When all bets are lost, we have a total loss of . The probability that the gambler does not lose all n bets is (1-). Then, the expected profit per round is . Since, clearly, B > 0, we get a negative expected profit if q > 0.5. The larger n becomes, the lower our expected profit is. So say the probability of losing is 0.6 and we do 9 bets where we start with 1 USD, our expected profit is 1(1 – ) = -4.159. So we can conclude you should most definitely not apply the martingale betting technique on bets where the odds are against you.

### Conclusion

The martingale betting technique, intuitively, seems like a technique in which you’ll barely ever lose. But taking a closer look at it, we see that it is not as solid as you might think. The event of a long losing streak is way more likely than most of us think, in particular if the total amount of bets we place increases. As I have also shown, the probability of winning using the martingale betting technique becomes even smaller if we bet on something where the odds are against you. If you do not want to lose any money, the best thing to do is to simply not gamble. But, if you can keep yourself under control, it’s never wrong to try your luck!

Dit artikel is geschreven door Lars Beute

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