Why is it so hard to win?

Many people, myself included, have often wondered why it seems so hard to hit a win 5, 20 or even 100 times in a row. Is it reasonable to practically never get ten wins in a row when the chance of winning is 10%, 25% or even 50%? Why can you play a game at a multiplier of 100 for 200 turns without winning? Is the game rigged, or is there a perfectly sound explanation?

Here I’m going to go through a few scenarios using a couple of in-house games you can find at most casinos. Let’s go through the likelihood of hitting a win in x amount of runs. I will also go through the math behind it and show how it’s calculated, giving everyone the tools to calculate the odds themselves.

I played Limbo at a multiplier of 100 but didn’t win even once

The chance of winning at a multiplier of 100x at Limbo (as well as classic dice, hash dice and ultimate dice) is 0.99%. But shouldn’t that mean you should hit 100x roughly every 100 turns?

Statistically speaking, you would hit a win for every 101 turns you play the game, but that is just a statistical measure. To illustrate, I played 10,000 games of Limbo at a multiplier of 100. Below are the first eight sets of 101 turns and the last game.

Betting statistics
Statistics from Limbo
Betting statistics
Final result from 10100 games of Limbo

During my 10,100 games (I played 101 at a time because statistically, I should get 100x once every 101 games), I could play several hundreds of games without getting 100x even once and then get 100x 3-4 times in another set of 101 games. The first ~400 games didn’t look good, showing a hit rate between 0.0 – 0.24%. But looking at the last live stats screenshot, the total hit rate is up to 0.9%, which is very close to the 0.99% that we statistically should get.

I want to show that it is perfectly normal to play 100, 200 or 500 games without hitting 100x. The statistical measure should be interpreted as an average, and the more games you play, the closer to that average you will be. Had I played 100,000 games of Limbo instead of 10,000, the total percentage would have been even closer to 0.99%.

For context, let’s see the odds of losing 100 games in a row. Every game has a 99.01% risk for loss.

99.01 * (0.9001^99) = 36.97

The odds of losing 100 games in a row at a multiplier of 100 is 36.97%

Then what about Coin Flip, which has a 50% chance on each turn?

I won’t play 10,000 rounds for this game because I would die of old age before it’s done.

A coin flip has a 50/50 chance to fall with heads up. What are the chances of the coin falling with heads up five times in a row, or fifty?

The chance of getting heads (or tail) x consecutive times in a row is calculated very similarly to getting the ball to go left x consecutive times in the game of Plinko.

Flip nrChance
150%
225%
312.5%
46.25%
53.125%
*
*
*
500.000000000000089%

To calculate the odds of getting fifty heads or tails in a row, we use this:  50 * (0.5^49)

But I’ve played 50 games without getting five heads in a row

There is an equation for calculating the number of coin flips you statistically have to do to get five heads or tails in a row, which looks as follows.

Picture explaining equation

е= expected number of flips. The expected number of flips to get five in a row is 62. To calculate how many flips are expected to get six in a row, you add the following after the part that represents 3.125%:

Equation

And so it continues with 7, 8, 9 etc. However, if you aren’t good at math, it is probably better to google search for something like “getting x coin flips in a row”.

And just as with the Limbo example, this doesn’t mean you are guaranteed five in a row because you play 62 games. It is just a statistical expression, and you might just as well have to play 150 games before you get five in a row. Or you could get five in a row on your first try.

Should you play 10,000 games of coin flip and calculate the average amount of flips it took to get five in a row, you would find out that it would be very close to 62.

When playing the Wheel, it says “Chance 1/30”, but I rarely win one of thirty games!

One chance in 30 games is the same as a 3.33% chance for a win. Statistically, you would hit a win once every ~30 spins. In 300 spins, I should get ten wins, statistically. Let’s see how it goes!

Betting statistics
Betting statistics

I was lucky initially; I got 13 wins out of 300 spins. That’s slightly higher than the stated 3.33% chance. But here, just as in the Limbo example, it starts to even out towards the end. Now I only did 300 spins; had I done 3,000 or 30,000, my profit would have been close to zero.

One way of looking at things when you don’t win as often as you think you should is to ask yourself what the risk is for losing on each turn rather than the chances of winning. Here we have a 96.67% risk of loss for every spin, so let’s calculate how big of a risk there is to lose 100 spins in a row.

96.67 * (0.9667^99) = 3.38%

Statistically, it’s about the same risk to lose 100 spins in a row as it is to win one spin out of 30! When estimating the risk versus the chance of winning, that’s something to consider.

Conclusion

Probability doesn’t work the way we intuitively feel that it should. You always have to take all the previous rolls into the calculations when calculating the probability of a series of events. 10,000, 50, and 300 rounds have been used in the above examples. But it can easily be changed to 3, 12 or 500 by adding multipliers. For example, the chance of winning two times in a row at a 20% chance would look like this:

20 * 0.2 * 0.2 = 0.8%

And 10 times in a row would look like this:

20 * 0.2 * 0.2 * 0.2 * 0.2 * 0.2 * 0.2 * 0.2 * 0.2 * 0.2 = 0.00001024%

The proper mathematical way of writing this is: 20 * (0.2^9)

Tip: You can copy and paste calculations such as 96.67 * (0.9667^99) in google, and it will show you the result!

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