Repeated bets and dependence

Add means freely, but inspect covariance before adding variances.

12 minutes · intermediate · Original Poker4Quant lesson

What you will learn

Compare independent and linked trials with the same one-trial distribution.

Prerequisites: Variance and standard deviation · Equally likely outcomes

Same marginal distributions, different totals

For two payoffs, Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y). Independent finite-variance payoffs have zero covariance, so their variances add. Equal one-trial means and variances do not establish independence.

Two fair ±10 payoffs have variance 100 each. If independent, total variance is 200. If the second always equals the first, the total is ±20 and variance is 400. If the second is always the negative of the first, the total is zero and variance is zero.

A losing streak does not force compensation

In independent identical trials, observing a past loss does not change the next trial’s distribution. A claim that the next trial “must be due” for a win introduces a dependence that was not in the model. In real systems, dependence can exist, so it must be investigated rather than assumed absent.

For n independent identical finite-variance payoffs, total mean is nμ and total variance is nσ². The average payoff’s variance is σ²/n. This describes concentration of the average under that model; it does not eliminate the possibility of a negative total or permit unlimited stakes.

Worked example

Each of two independent rounds pays +30 or −10 with equal probability. Describe the total.

The four equally likely totals are +60, +20, +20 and −20.
Their mean is (60+20+20−20)/4=20.
Squared deviations from 20 are 1600, 0, 0 and 1600.
Their average is 800, equal to two times the one-round variance of 400.

The expected total is +20 with variance 800. One of the four paths still loses 20, despite positive expectation.

Common mistakes

Adding variances without checking covariance.
Believing a streak forces the next independent result.
Confusing a lower variance of the average with zero risk of the total.

Keep these ideas

Means add under dependence too.
Variance of a sum includes covariance.
Independent trials do not remember the previous result.