May 04 1 Response

What A Coin Flip Can Teach You About Expected Value vs Probability

For whatever reason, it took me forever to understand the difference between probability and expected value. I searched the web high and low, and as far as I could tell, there wasn’t a good explanation to be found. I spent way too much time trying to wrap my head around the concepts, and I hope my explanation below will shortcut the process for you.

First, let’s start off with some quick definitions:

Probability measures how certain we are a particular event will happen in a specific instance.

Expected Value represents the average outcome of a series of random events with identical odds being repeated over a long period of time.

Take a coin flip. Every time a coin is flipped, the probability of it landing on either heads or tails is 50%. To determine the expected value, we have to apply some numbers to the outcomes.

In a scenario where every time the coin comes up heads, you win $2, and every time the coin comes up tails, you pay $1, your expected value is $0.50 per flip. That is, if you flipped the coin twice, one time it will come up tails and you’ll pay $1 and one time it will come up heads and you’ll get paid $2. So after two coin flips, you have made $1 ($2 – $1) or $0.50 per flip.

Of course, we all know if you only flip a coin twice, you could easily end up with heads twice or tails twice. Fortunately, we can use math to show us the probability of a particular outcome occurring. If we wanted to know the probability of flipping a coin twice and heads coming up at least once, we would apply this formula:

1 – (number of non-desired outcomes divided by total number of possible outcomes) ^n where n is the number of times a particular event is being repeated.
For the above example of a coin being flipped twice, it would be 1 – (1/2)^2 or 75%.

  • 25% of the time, you lose twice, Tails/Tails
  • 25% of the time, you win twice, Heads/Heads
  • 50% of the time, you win once Tails/Heads or Heads/Tails

Now, read the above again, and realize, it has NOTHING AT ALL TO DO WITH EXPECTED VALUE. It’s a straw man argument to try and correlate the two different concepts. It’s impossible to win $0.50 on any flip (you either win $2 or lose $1), and 25% of the time you won’t win anything at all. It is easy to associate the two concepts with each other, but don’t. You need probability to get expected value, but that’s it. There are no more math principals to imply. Here’s another example:

A Weather Example
Take a 5-Day Weather forecast where there is a 20% chance of rain every day. The probability of it raining at all during those 5 days is 1 – (4/5)^5 or 67%. Every day, there is 1 outcome we want, rain (20%) and 4 outcomes we don’t want, no rain (80%). That’s where the 4/5 comes from. And 5 represents the number of days we are repeating this particular event.

So if we were gambling, how should we look at the outcome? Should we focus on the 67% of rain over the course of 5 days or the 20% chance of rain each day? It depends completely on what the bet is. “Will It Rain This Week” is a very different bet than “Will It Rain Today”.

In “Will It Rain Today” bet, if we have $5 to wager, and we spread it out over the week, betting $1 every day for it to rain against $5 that it doesn’t, the expected value is $0.20 per day. On the day it rains you’ll win $5, and on the four days it doesn’t, you lose $4. That means every day, you’re expected value is: $0.20.

The “Will It Rain This Week Bet” is a completely different bet with completely different odds, and you’ll do nothing but confuse yourself if you look for links between the two bets. In this instance, we bet $5 against $5 as a one time bet and our expected value is $1.67 (2/3 of the time you win $5; 1/3 of the time you lose $5).

Using the rule of large numbers, over a long enough period of time with the above odds, we will definitely make money or in gambling terms have a positive expected value (EV+). This is the basic principle that keeps the lights on in Las Vegas; only it’s the house with with EV+ where you, the gambler, will have an EV-.

Expected Value (EV) gives you the expected return on a single event and cannot be correlated to several events, that’s not what it does. In the “Will It Rain Today” example above, your EV is $0.20 on every bet. It is the same on the next bet and the next one. If the chance of rain were 20% on three days, your expected value on those days is still, $0.20. It is not an additive principal. If the, Probability of a specific event changes, your Expected Value will also change – and this is the only time it will change.

Just because you have a 1/5 chance over 5 days does not mean an event will happen. In fact, we can show definitely, it will only happen 2/3 of the time. On the other hand, there will be weeks where it will rain multiple times as well. Over the long run, you can expect it to rain 1 out of every 5 days, but you can never say with certainty when that 1 day will come. There will be stretches of no rain and stretches of rain. Over time, though, there will be a regression to the mean. This is why you can bank on making 20 cents every day. This is expected value.

Probability of Ruin
Am I telling you to always take the risk if you have a positive expected value? No, there is one more principle to consider called risk of ruin. It takes into account the probability of winning, the probability of losing, and the portion of your finances at risk. This isn’t a hard and fast number, but more of a personal risk tolerance equation. While some people would be perfectly happy risking all of their assets for a 2/3 chance of being able make enough money to retire, others would focus much more on the 1/3 of the time you ruin your finances.

All of these concepts can be applied any time you are dealing with risking money based on probabilities. In this sense, gambling is the same as trading stocks or trying to determine if you should pursue a specific business opportunity.

Good luck, and I hope this helps someone.

One Response to What A Coin Flip Can Teach You About Expected Value vs Probability
  1. John Parker

    Hey, Greg,

    Congrats on Buzzam Radio. I want to check it out. Is it available for Android phones? And if so, where can I download it?

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