# Gambling with the Martingale Strategy – Numberphile

We’re gonna go gambling. We’re creating a- let’s call it a fictional casino and we’re playing a game with a 50/50 chance of winning. Okay, so we’re doing roulette – red or black so there’s no zero, so it is actually a 50/50 chance red or black. What we’re going to look at is something called the Martingale Strategy. Every time you lose a bet you double down on the next bet; and therefore eventually you will win – is the idea. – (Brad: What do you mean by) (double down?) So if you bet one pound and lose then you bet again and you would bet two pounds; and then if you lost again then you go four pounds; if you lose you go eight pounds. So you double the last bet you made. Provided you have pretty much unlimited money you will eventually win.

A win is: come out of our fictional casino with more money than I went in. (You want to make a profit.) – Exactly we’re making a profit. – (Should we say we’re) (always betting on red?) (Or black or-?) – I think- I’m tempted to go black. So I write with a black pen. I think if I was given the choice I would- I would I’d like a- I’m going for black over red I think. (Okay so we’re betting on black.) – Betting on black. And we’re going to keep going; you think about well when is the first time I lose? So if I lose at step k I’ve had my bet at step 1, and I bet one pound, and then I must have lost because we’re assuming we’re going to keep going until this number k. Then at step 2 I bet two pounds. Then step three – again we’re assuming I’ve lost at step 2 – I double my bet so this is now going to be four pounds. Step 4 is going to be double that, eight pounds, and this carries on. – (If you win your money gets) (doubled?) – If I win my money will be doubled, yes, it’s a 50/50 game such that whatever I bet- like in, exactly like roulette. We don’t have a zero but the rules otherwise are the same. If I bet on black, which is what we’re doing, if it lands on black I double my money. (Because that zero is how casino is insured.) The zero- exactly. You know this way. The zero is how- casinos know this and this is why they have zero, because it’s never a 50/50 chance. But our mathematically fair casino gives us a true 50/50 chance. At step k then we can figure out how much we would be betting because there’s a pattern here. So at step 1 I bet one pounds, step 2 two pounds, step 3 four pounds. So what’s happening with our bets is these are increasing powers of 2. So this is times 2, times 2, times 2. So the actual formula is going to be 2 to the power of k minus 1. So we can just check this: so at step 1, so k is 1; 2 to the power of 0 is 1. Step 2 k is 2, 2 to the power 1 is 2; 2 squared is 4, 2 cubed is 8 so at step k we have just bet 2 to the k minus 1 pounds. If I then lose at step k my total loss so far is the sum of all of these bets; so 1 plus 2 plus 4 plus 8 all the way up to 2 to the k minus 1. We can write this as the sum from j equals 0 to k minus 1 of 2 power j. So 2 to the 0 is 1 then plus 2 to the 1 is 2 plus 2 squared is 4. So this is a shorthand way of writing out this sum. And this is what we call a geometric series; we have a formula for the sum of a geometric series so we can plug it in. It’s the first term, which is 1, multiplied by 1 minus the next term in the sequence. So it ends at 2 to the k minus 1 so if we were to go to the next step that would be 2 to the k; and then we divide by 1 minus the common ratio between the terms. So what we’re multiplying by at each step, so divided by 1 minus 2. So you tidy this up and this is 2 to the k minus 1. So this is the total amount of money I have lost when I lose my bet at step k. So in the next step I’m going to bet 2 to the k pounds – I’m going to win hopefully and then that covers my losses with one pound to spare. So after all of this effort, after losing at some step if I follow the Martingale strategy I will eventually win – we hope – before I run out of money, that’s a whole other question, and I will win one pound. So we’re not exactly- like it’s good, it’s a profit, but it’s it’s not like- we want more than a pound don’t we really? Like you don’t really go to a casino to make a pound. (But if you scaled it to 10 pounds or a) (million pounds as) (as your- like your multiple) (you you’re guaranteed to win a million pounds!) Yeah, so so to win a million pounds you’ve got to go through this strategy one million times, winning one pound each time. (Or just make that a million pound bet;) (a two million pound bet; a four million) (pound bet, all the way to a two) (to the minus 1 times a million-) – We could also do that yeah. So we could also change the amount we bet at the beginning and then that would be- the amount that you your initial stake is the amount that you’ll win through this strategy. So this strategy is going to win you, through this process, we’re going to win a pound each time. Now, there are problems with this. There is a reason that this is not a foolproof gambling strategy. First of all most games aren’t 50/50; if we’re doing roulette there’s a zero which is green, so it’s not red or black, so you’ve not actually got a 50/50 chance.

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