Variance, Volatility and Expected Value Explained
Variance, Volatility and Expected Value
Variance and Volatility are both very important concepts we should take into account when playing any game with any kind of mathematical probability involved. This can be a simple coin toss, rolling of a dice poker or pretty much any casino game. We cannot know the outcome of any game and sometimes we just know how probable it is in the long run.
Both of these concepts are very similar and while Variance is more of a mathematical concept, Volatility is more of a description of a measurement of fluctuations. Both are very similar and there is really no point in going into too much detail and we can just do with Variance.
Basically, Variance is a mathematical concept which measures how far a set of random numbers are spread from their mean. This might sound difficult, but bear with us, it's actually really simple.
Variance in Poker and Casino Games in Practice
Let's explain Variance on a simple coin toss. Say we toss a coin just once. There is a 50% chance we hit heads and a 50% chance we hit tails. In the long run, it is expected that both heads and tails are hit with the same frequency, but the reality may very well differ. If we toss a coin only one, there is only one outcome, which will differ from the long run theory, which is a 50:50 frequency. If we toss a coin twice, there is a 50% chance that we hit the same side of that coin both times. If we toss that coin a million times, it is almost impossible to only hit heads (or tails).
In reality, after a million coin tosses, the frequency of heads and tails will be pretty close to that 50:50, as it should be. However, there is always a chance, even if extremely slim, that we will only hit one side of the coin for a million tosses. This would be very far from our expectation, which is that we hit both sides with the same frequency.
The farther it is from the long-run expectation, the less probable it is with large sample. There is always a chance, but very slim. It is actually so slim that even if you have been tossing a coin for the entire duration of the existence of the universe, we would not achieve even 1,000 consecutive heads hits, let alone a million. In practice, with large enough samples, the factor of (bad) luck will disperse and the results will balance each other out. Short run results however may be very different from the expectations.
The distribution might look like this. Disregard the exact numbers, but it is most probable that the results are near our expectations, and the farther from the middle of the function, the less probable it is that the results will be in that section
Expected Value (EV) basically tells us how much money we expect to be making with each result. Say we are tossing a coin and every heads will earn us £1 and every tails will cost us £1. We expect to be making £1 50% of the time and losing £1 50% of the time. We expect to be making exactly £0, whatever the actual result.
The equation for EV is very simple:
|EV = (probability of winning * what we can win) - (probability of losing * what we can lose)|
Now if we substitute, we are given EV = (0.5 * 1) - (0.5 * 1) = 0
Now what if we win £1 with either heads or tails and never lose anything? Is is quite clear that the EV = 1 and we are actually playing a zero-variance game. There are no other results other than our victory and if we tossed a coin a million times, we would make £1,000,000 with a 100% certainty.
Now let's change the rules a bit and let's play with a dice with a hundred sides. Only one side wins us £100, other 99 sides don't win or lose us anything. If we substitute, we get the equation EV = (0,01 * 100) - (0 * 0) = 1
The EV is exactly the same as with the £1 win with every single toss, but it is quite clear that something is very different.
Why Are Those Two Games Different in Regards of Variance?
The answer is that those two games are different because of the variance. In the long run, we will be making the same money with every single toss and we are expected to hit that winning side once every 100 rolls winning us £100, but we can be rolling that dice for a very long time and never actually win anything, and other times we can hit that lucky side very frequently.
Variance makes it so that we need a huge sample size to balance the results, while the game with tossing a coin makes us very stable money, and actually the same money in the long run.
If we want play responsibly, we would always choose a low-variance (or zero-variance in this case) option, if the EV is exactly the same.
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How Is This Important for Casino Games and Poker?
Variance and EV are very important when we choose the game we want to play and especially when deciding what we want to achieve. There are hundreds and hundreds of slot machines with very different Variance and Volatility. Some slots hit some winning combination with almost every spin, but the winnings are smallish and may not even pay for the cost of that spin. Some slot machines have very infrequent hits, but when they actually do happen, they are for a lot. The thing is that every slot machine has negative EV for every player and playing them is a losing venture in the long run.
It might still be ok to play them, because while the game itself may not be profitable, we are usually given come sort of cash bonus or free spins, which may balance the scales.
Should We Play High or Low Variance Slots?
There is another reason why we should take Variance and Volatility into account. If we are playing a low-variance slot machines and are expected to lose over time, we can be very sure that we will slowly be losing our bankroll. If we just want to have fun and consider slot machines a hobby without any aspirations to make any money, these slot machines are just perfect.
Some other slots allow us to win even millions or tens of millions of pounds in one spin. These are also not profitable in the long run, but we can still play and hope to luck out. It's the same as if we played with a million-sided dice and only one side won us that million pound Jackpot, while all of the other sides lost us only £1 each. If we hit that 1 million pound Jackpot on our 500,000th roll, we are £500,000 in profit! Yes, we needed to be lucky and it could have taken us many many rolls, but we are giving ourselves a chance to actually get rich by being on the good side of Variance.
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