A fair coin is equally likely to get a heads or a tails in every flip, right ?

So, for a fair coin :

Probability (of getting Heads) = 1/2 and Probability (of getting Tails) = 1/2

**Ten Flips**

But this 1/2 and 1/2 is a long term expected value. If you flip a coin 10 times, you might get seven heads, and three tails, which is not 1/2 and 1/2

In this case

**P(Heads) = 7/10 and P(Tails) = 3/10**

So, what does the premise of 1/2 and 1/2 mean ?

It means, rather than flipping a coin ten times, if you flip the same coin a thousand times, or a million times, **approximately** 50% of those flips would be heads and the other 50% is likely to be tails

Great, let us do a couple of test cases

**Case 1 : 1,000 Flips**

Say, for a thousand flips, you get 550 heads and 450 tails

P(Heads) = 550/1,000 = 5.5/10

P(Tails) = 450/1,000 = 4.5/10

If you compare this result to the one we got for the ten flips above,

P(Heads) changes from 7/10 to 5.5/10 (getting closer to the long term expected 5/10)

P(Tails) changes from 3/10 to 4.5/10 (getting closer to the long term expected 5/10)

**Case 2 : Million Flips**

For a million flips, you may end up with 490,000 heads and 510,000 tails

In this case,

P(Heads) = 490,000 / 1,000,000 = 4.9/10

P(Tails) = 510,000 / 1,000,000 = 5.1/10

Let us compare this result to the 1,000 flips case above,

P(Heads) changes from 5.5/10 to 4.9/10 (getting closer to the expected 5/10, or 1/2)

P(Tails) changes from 4.5/10 to 5.1/10 (getting closer to the expected 5/10, or 1/2)

**Perfect, so what does this mean ? **

**It means, as you flip the coin for an infinite number of times, the number of heads and tails you get - will get closer and closer to the mathematically expected 1/2 (for a fair coin). In more mathematical speak, the expected value would tend to the true value (1/2 and 1/2) with increasing number of coin flips**

Great

But why do we care about this ? Because this is one of primary reasons why Gamblers go broke

~ we call this problem, The Gambler’s Ruin. The very phrase has the ring of a Shakespearean tragedy to it, don’t it ?

So, imagine this simple game, you and I play a game and we both start with $ 100. We flip a fair coin - if we get a heads, I give you $ 10, and if we get a tails, you give me $ 10. We both start with 100 bucks, so depending on whether we get heads or tails, our winnings might go up and down for a while before one of us ends up losing

Small stakes, typical night, no stress

But now, imagine if you play the same game with the Bellagio or Caesars Palace in Las Vegas. Same thing, heads means the casino gives you $ 10, and tails means, you give the casino $ 10

**So, what’s the difference ? or is there even a difference ?**

When you and I played, you and I had $ 100 each. But when you play against the Bellagio, you started with $ 100, and the Bellagio started with infinite amounts of cash. The Bellagio literally has millions to bet and possibly lose … until they start winning. They can wait - for the odds to turn in their favor

That makes all the difference. This is why most untrained gamblers go broke against Las Vegas

Imagine this : you win $ 10, when we get a heads, right ?

You start to play, and for some god-forsaken and mathematically cursed reason, the first ten flips are tails. The chances of this happening are (1/2)^10 = 1 in 1024, but it can happen. Since you got ten tails, you have to give over ($ 10 for each tail x 10 tails) $ 100 to the casino, and now you are done

You are bankrupt

But, if you play ten more times, the next ten can be heads, and out of the 20 times, we got ten heads and ten tails. So, the 1/2 heads and 1/2 tails expected value still holds, but the only tragedy is, you couldn’t take advantage of it because you are already toast after ten flips

Now, imagine the roles to be reversed

If the first ten flips are heads, you win $ 100 from the Bellagio. You are happy and you just ordered yourself another Bloody Mary

But the Bellagio isn’t going to stop (unless, you walk away from the table). The Bellagio has practically infinite amounts of money to bet, and they will wait until the luck of the draw turns against you. They’ll wait until multiple tails start to show up, so they can recover their loss, and then some … and then some more

They have time to wait until you end up like Nicolas Cage in Leaving Las Vegas

**To bring this home : the bigger the financial advantage the casino has over you, the more likely you’ll eventually go bankrupt**

in other words ~ the tragically poetic Gambler’s Ruin

**It’s a beautiful probability problem **

Last week we talked about recursive probabilities, Gambler’s Ruin is another type of recursive probability problem, and we typically solve it using Markov chains

Here is the mathematical intuition of it

If we define Pi (where i is the $ you have in your hand currently), the probability of you winning N dollars in the end is essentially (i/N), when both outcomes (heads and tails) are equally likely

If you have $ 50 in hand, the probability of you winning $ 500 by the end of the night is

P50, for a N of 500 = 50/500, or 1/10

Say, you lose the next three rounds because you got tails the next three times. You have to give over $ 30 to the Bellagio, and now you have ($ 50 - $ 30) $ 20 left

The likelihood of you winning $ 500, when you have $ 20 in hand now, is 20/500, or 2/50, or 1/25

**Just like that - your odds went down from 1/10 to 1/25 (to win $ 500 by the end of the night)**

Say you get one more tails, and the money you have goes down to $ 10. Now your odds of getting to $ 500 is i/N, 10/500, which is 1/50

Once you start losing, your odds of getting to your target winnings ($ 500) get longer and longer

~ in the end, you’ll be ruined

You don’t have infinite amounts to bet and buy time till the luck of the draw turns in your favor … but the Bellagio does - and that is the entire ball game

**Reference :** Gambler’s Ruin : Columbia University, MIT and Towards Data Science

I remember as kid "discovering" a foolproof way to make money gambling (about the same age as one discovered how not to lose at Tix Tax Toe). <Bet, if win, stop, if lose double stake, bet>. But I also realized that 1) I was unlikely to find anyone to the other side of these bets and 2) that I might not have enough money to wait long enough to win.