Gambler’s Ruin – Life as a Coin Toss

“You can do a lot in a lifetime if you don’t burn out too fast.” – Rush

(Photo: Pauli Antero)

One of the most common mistakes I see entrepreneurs and everyday people make, is not building up their cash fund and assuming that the present economic conditions will continue into the future. In business I often see companies who over expand and go under once the economy or their business takes a turn for the worse. They fail due to lack of reserves needed to weather the storm.

I often see people who get a raise and right away start increasing their spending instead of increasing their savings. Once hard times hit they find themselves a paycheck away from losing their house and everything they worked for because they failed to save and assumed that the current good times will last forever.

To demonstrate this I want to introduce the concept of “Gambler’s Ruin”. It’s also part of the reason why Las Vegas exists. Let’s say tomorrow a very rich man comes to you and says he wants to have a coin flipping contest. Heads he gives you a dollar, tails you give him a dollar. You think about it and see that since your odds are 50/50 there’s really no point of you playing this game. The rich man then announces that he would sweeten the pot for you. He makes the following rules:

  • You will play for $1 per coin toss but all you can put in is $5 and no more, so the most you can lose is $5 when the game ends, on the other hand the most he can lose is $1,000,000.

  • He will use a trick coin where heads comes up 51% of the time, meaning that the trick coin is in your favor.

  • He will use 1,000,000 coin tosses.

You think about the odds; since statistically on average heads should come up 51 times out of 100 throws means that on average you stand to make $2 with each 100 tosses ($51 you win if heads – $49 you lose if tails = $2). Since the game will go on for 1,000,000 tosses, you would stand to gain, on average, $20,000 ($2 gain per 100 tosses = $20,000 for 1,000,000 tosses). You think about it and see that this is a great deal so you decide to play! Was this really a good idea?

The answer is maybe, maybe not. It is almost a given (about 82% chance) that you would lose. Why? Because sooner or later the coin toss would turn against you and you would lose your $5. Once you do the game is over since the rules stipulated that you can’t put in more than $5 while the rich man can keep playing and playing even if he’s behind for a while.

(Geek Alert: I show all work below)

So even if the odds are deeply in your favor you would still lose due to lack of funds and not being able to continue playing if the randomness of the coin turns against you. (This is also why in Vegas the house usually always wins, not only are the odds stacked against you but eventually you run out of money while waiting for the odds to turn in your favor. The casino can stay solvent much longer than you.)

The same concept is mirrored in life, even if you have a good money making system or a good income. Without a good money cushion, which you need to be protected, you are very likely to lose. The rules are the same in business and finance; once you are broke it is game over.

To me true financial wealth is not measured by what you have but what you could have if you wanted it. In other words the only money that counts are liquid assets not your cars, swimming pools or designer clothes. Since most everything you buy loses at least 50% of its value if you had to sell it. It’s crucially important to have a financial cushion in order to be protected against any harsh surprises which may (and will) come your way. You cannot control life’s flip of the coin but you can protect yourself to make sure that you live to play another day.

Michael Page

Geek Notes:
How to calculate your chance of going broke in a 51% coin toss with 1,000,000 tosses if you start with $5. (Gambler’s Ruin)

The number of played games is 1 million. So the possibility of being ruined goes like this
n1 = 5 – the money the player 1 has
n2 = 1 000 000 – the money the player 2 has
P1 – the possibility of Player 1 being ruined
P2 – the possibility of Player 2 being ruined
p = 0.51 (51%) – Player 1 wins with each flip  with probability p
q = 1-p = 0.49 (49%) – Player 2 wins each flip  with probability q
The formula for probability of each ending penniless is:

We need to calculate  
Lets just calculate
Let t =
We use the natural logarithm function on both side so we have:
log t = log (
log t =
log t = -17374. 09606942270441076528039818
Lets use the inverse function of log (that is the definition of natural logarithm):
t =
t =
Lets get back to our formula:
P1 =
P1 =
P1 = 0,8187089153077668509543378319287

P2 =

=  1- 0,8187089153077668509543378319287
P2 = 0,1812910846922331490456621680713 or 82%

If you made it this far and understand it, congratulations you are officially a geek. You can now go back to scouring  Ebay for more stuff to add to your Dr. Who Memorabilia collection.


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