Concept #4: Mathematical Edge

Flipping a coin is an easy game to play, but there’s a valuable probability lesson that comes from the game. We know when flipping a coin there are 2 possible outcomes Heads and Tails. Both of which are equal. And, “Equal” means that each outcome has a 50% chance of coming up.

But what exactly does that mean?  Over the near AND long-term.

Well, over the long-term with continuous flipping, if you added up all the times the coin came up heads and tails, then you would get results that approach 50% for each outcome. However, at any given time, especially in the near term it may not be 50%, sometimes not even close to it!

Understanding this, let’s play a game - FOR MONEY of course!

The Game is HEADS OR TAILS with 3 simple rules:

Would you play this game?

Most of you will answer: “Of course.”
But, I want you to be able to understand "Why?"

When playing a probability game, you want the EXPECTATION of GAINS to be greater than the EXPECTATION of LOSSES.

Let’s discuss this further.

Expected GAINS can be calculated by taking the probability of a positive outcome multiplied by the potential Profit or gain in that outcome.  And the expected LOSSES are calculated by taking the probability of a negative outcome multiplied by the loss in that outcome.  Two things to note: the probabilities should add to 100%; and, there can be more than 2 possible outcomes.

So, let’s calculate the expected gains and losses in this proposed game.

The Probability of a positive outcome is 50% where you win $1.20.  The Probability of a negative outcome is 50% where you lose $1. Multiplying each side results in an Expected Gain of $0.60 and an Expected Loss of $0.50.

The Expected Gain is greater than the Expected Loss. This means that it’s a probability game you should play. The expectations of success are in your favor.

In fact, when the Expected Gains are greater than Expected Losses then you have what is called: Mathematical Edge.This means that in the long run with continuous play, the math will work itself out and you’ll profit with certainty.

Going back to our example.
What if we changed the payout?
Instead of losing $1.00, you lose $1.20. And instead of winning $1.20, you only win $1.00.

Here’s the calculation.

The Probability of a positive and negative outcome is still 50% but the gain is now $1.00 and the loss is $1.20.  The results are reversed with the Expected Gains equal to $0.50 and the expected losses equal to $0.60, making the expected losses higher than the expected gains.

With these payouts, you don’t want to play this game. There is NO Mathematical Edge.

Does Mathematical Edge Guarantee Profit?

Anything can happen in the short-term. Playing a game of Heads or Tails and flipping a coin, regardless of the payouts and expectations, can result in losses in the short term. However, over the long term and with greater flips, Mathematical Edge wins out!