Expected value is the probability-weighted value of all possible outcomes of a random variable. In gambling or investing, E(V) includes the monetary value of each outcome. Expected growth takes the same variables (outcome, probability, and payoff), and includes the effect of compounding.

  • Expected value and expected growth will never appear as an actual result. E(V) is the weighted mean over an infinite number of trials, while E(G) indicates the likely trend, not a final outcome.
  • Expected value will almost always be greater than expected growth. (Where probabilities are 0% and 100%, E(V) and E(G) will be equal);
  • It’s possible to have a positive expected value, but a negative expected growth; and
  • Over-betting on expected outcomes will always have more negative long-run consequences than under-betting.

What is expected value?

Expected value is the sum of the weighted products of the possible outcomes times the probability of those outcomes. In gambling or investing, those products are multiplied by the payoff of each outcome to find an expected value in dollars.

Imagine a fair six-sided die, but instead of a number on each side, there’s a letter. Next, imagine that the game is to bet on what letter will come up. If you roll your letter, you win 6x your bet.

You bet on F. What’s the expected value of this bet?

  • What are all the possible outcomes? A, B, C, D, E, and F.
  • What is the probability of each outcome? All are equal: 1/6 or 16.7% or 0.167.
  • What is the payoff of each outcome? What is each outcome worth?
  • A through E are worth -1. You’ll lose your bet.
  • F is worth 6. You’ll win six times your bet if F appears.

So, what’s the expected value of the bet?

E(V) = -1(\frac{1}{6}) -1(\frac{1}{6}) -1(\frac{1}{6}) -1(\frac{1}{6}) -1(\frac{1}{6}) + 6(\frac{1}{6})

E(V) = -1(\frac{5}{6}) + 6(\frac{1}{6})

E(V) = -\frac{5}{6} + \frac{6}{6} = \frac{1}{6} = 0.167

So how much do you want to bet on each roll?

It’s tempting to think that if the expected value is 0.167 that you should bet 16.7% of your bankroll. But that would be way too much.

With that bet, the expected growth is negative. The long-run expected value of betting 16.7% of your bankroll on each roll would be $0.

What is expected growth?

The shortcoming of expected value is that it doesn’t consider the effect of compounding over a large number of trials, especially the effects of negative compounding. That’s where expected growth comes in.

The formula for expected growth is:

E(G) = (1 + b * f)^{np}(1 - f)^{nq} - 1, where

  • 1 represents your bankroll;
  • b is the odds of the bet;
  • f is the portion of our bankroll at stake;
  • n is the number of trials;
  • p is the probability of a winning outcome; and
  • q is the possibility of a losing outcome (aka. 1 – p).

So for our letter-sided die above, the expected value is 0.167. What is the expected growth if we bet a portion of our roll equal to the expected value of 16.7%?

  • b is 6. We’ll win six times our bet if “F” comes up;
  • f is 16.7% of our bankroll;
  • n is 1 for betting just once;
  • p is 0.167. There’s a 1/6 chance that we’ll win; and
  • q is 0.833 (aka. 1 – p). There’s a 5/6 chance that we’ll lose.

So the expected growth would be a factor of 0.9644.

E(G) = (1 + 6 * 0.167)^{1(0.167)}(1 - 0.167)^{1(0.833)} - 1

E(G) = 0.9644

Each time we play, our bankroll is expected to become 96.44% of what it was. If we start with $100, then after one roll, we’re expected to end up with $96.44.

That was on one roll. But if we keep betting like this, it’s even worse.

How does expected growth include compounding?

The expected growth above was based on betting just once. If we continue to play this game, then the factor we came up with is multiplied with each roll.

If we have the 1-roll growth factor, we can still find out what the expected growth is on subsequent rolls. Over any number of rolls, we put the expected growth factor to the power of n, the number of rolls. So after 10 bets on this game, our bankroll is expected to shrink by a factor of 0.6957.

E(G) = 0.9644^{10} = 0.6957

If we play this game ten times, our bankroll is expected to be 69.57% of what it was. Starting with $100, we’re expected to end up with $69.57. And if we keep playing long enough, $0.

E(G) = 0.9644^\infty = 0

Will there be the occasional upward trend in our bankroll? Of course. Expected growth doesn’t predict each outcome. There will be “winning streaks”. But the long-run trend will be negative, eventually ruining us.

So is it the game that sucks or is it our betting strategy?

It’s our betting strategy. Our bet is too big.

The expected value is positive, so we do have an “edge”. But now we need to tailor our bet to the game so that the expected growth is also positive.

What if we bet 1% of our roll?

E(G) = (1 + 6 * 0.01)^{1(0.167)}(1 - 0.01)^{1(0.833)} - 1

E(G) = 1.0014

So betting only 1% of our bankroll would have a positive expected growth. Over a long enough timeline, our bankroll would grow rather than shrink. Over ten bets, our bankroll would grow by a factor of 1.014.

But making 1.4% over ten bets is a crappy rate of return. Can we do better without ruining ourselves?

How do we find the best bet?

This takes us back to the Kelly criterion. If we divide our edge by the odds on a winning outcome, we’ll have the optimum portion of our bankroll that we should bet on each roll. For our dice game, our best bet size would be 2.783% of our bankroll.

f = \frac{Edge}{Odds} = \frac{.0167}{6} = 0.02783

And the expected growth of betting 2.783% of our bankroll would be a factor of 1.0023.

E(G) = (1 + 6 * 0.02783)^{1(0.167)}(1 - 0.02783)^{1(0.833)} - 1

E(G) = 1.0023

Over ten rolls, our expected growth is a factor of 1.023.

E(G) = 1.0023^{10} = 1.023

It’s still not a great rate of return, but it’s the best we can expect with this game.

(The smart thing to do is to find a better game…)

Why should I care about expected growth versus expected value?

For long-term growth to be positive, bets need to be appropriately sized. Bigger is not always better, and expected value doesn’t paint the whole picture. If you only consider expected value, then it’s likely that you’ll over-bet.

If our bet is too big, that bet can turn a positive expected growth into a negative one. Repeated over enough trials, the eventual outcome is financial ruin.

Right-sizing bets, or under-sizing them when the variables can’t be known, is the only way to guard against negative expected growth.

For more detail and examples, check out the following two posts from the Sports Book Review forum: