Mathematical Expectation In Trading

I received a lot of emails to my post yesterday about the expected value of a trade. Only one person got the correct answer to the game of flipping 2 Aces off the top of the deck – what I’ll call the Game of Aces.

Now I can easily publish the correct answer, but that would rob you of an aha moment I believe, so I’m not going to do that.

The math behind playing the Roulette Wheel is why you’d never want to play it, despite the fact that you might get what economists call Economic Utility – pleasure or fun out of playing a game. For most serious traders though, losing money is not fun. Unless, of course, the trader gets more out of being a martyr or playing the role of victim, then it’s a different kind of game.

Anyway, to recap the Roulette Wheel, you have 38 spaces to bet on: numbers 1 through 36, a 00, and the blank space. It costs $1 to play and the winner pays $35.

Most of the answers I got from the Game of Aces, said “Yes, you play it all the time because you get $100 for your $1 bet, so that’s 100:1 – who wouldn’t want that all the time?” I think people get conditioned by the analysts on TV who suggest that “XYZ stock can go another $15 before we’d be worried.” They see the $15 as the payoff, but don’t see the downside risk.

What everyone left out was the frequency with which that result occurs. [For all the Lottery players, sit up straight and read closely.]

The payout ratio of the bet is irrelevant if the frequency with which it occurs is negligible. In the case of the Roulette Wheel, the payout ratio is in fact 35 to 1. However, the frequency (probability) with which you win per $1 bet is 1/38, or about 2.63% of the time. The other 97.37 % of the time you will expense your $1 bet. Playing this game with a fixed-budget constraint will surely make you Broke.

That’s why all the states in the Union market the living hell out of the Lottery as such a dreamy proposition: they sell you on what you could win, not the rarity of the occurrence. You’ll never win.

Spin the wheel 10,000 times (for $10,000 in bets), and on average you’ll get these results:

263 wins for $35 each = $9,210

9,737 losses of $1 each = ($9,737)

What’s compelling is not the 35:1 payout: it’s the frequency with which each occurs. So, if you’re lucky, and you come to the wheel and make $10,000 worth of $1 attempts, you’ll go home with $9,210 having lost $527 – the house’s edge. If you actually calculated the mathematical expectation from playing the Roulette Wheel, you’ll know it is – (1/19), which is -.0526 per attempt. Multiply that by 10,000 attempts and you’ll get approximately negative 527.

Think about this in the Game of Aces example from yesterday, and think about it next time you play Lotto. Don’t make a single trade with real money until you have an idea of how your thoughts manifest into winning and losing trades, and the frequency with which they occur. If you don’t care about this, you may be drawn to trading for the action, and not to make money.


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  • Zakir

    > Don’t make a single trade with real money until you have an idea of how your thoughts manifest into winning and losing trades, and the frequency with which they occur.

    Great advice. The tough part is automating this, or if not automating it, remaining cognizant and disciplined enough to execute your strategy. I’ve had winning strategies that have unfortunately shifted with market and account conditions (such as putting on too much risk when ahead), so a primary focus of mine is to incorporate these dynamics into my strategy instead of having a strategy that changes with the tides.

  • martinkronicle

    Hi Zakir – I agree, it’s hard without a computer to run through thousands of simulations to get to the exact number. Secondly, it’s hard for some to remain objective: a computer simulator will take all the trades and not amplify a big winner or conveniently “miss” a big loser. But even if you had the true expected value for your style of trading or system, it’s not predictive. It’s only there to give you an idea of what to expect – but I applaud you for doing the work. It’s leagues beyond a lot of crap I see.