The origin of Plinko

The game of Plinko has its origin sometime in the late 1800’s and was originally constructed by mathematician and statistician Francis Galton. He constructed what is called a “Galton box” with the purpose of proving that with a large enough sample, a binominal distribution (i.e., when there’s 2 options with equal chance of outcome) will result in a normal distribution (also known as the Bell curve).

If you were to collect every ball from a game of Plinko down at the bottom, you would find that the balls would form a near perfect Bell curve.

But why is that? Surely there must be some scam to all this? After all, there’s a 50/50 chance for the ball to go left or right at every step all the way to the bottom. Shouldn’t that mean that every area at the bottom has an equal chance of catching the ball? Nope, unfortunately that’s not how it works.

So how does it work?

Let’s say we only have 1 peg; the ball would have a 50% chance of going left and a 50% chance of going right.

But if you add another row with 2 pegs in it, the results will be a bit different. We first have an equal (50%) chance for the ball to go right or left. Let’s say our ball went left. Then we have a 50/50 chance of it going right or left again, but to get the probability of the ball going left and then left again we must take the whole board in to account.

Because there´s half of a 50% chance for the ball to go left twice, or two 25% chance of the ball going in the middle. For every row you add, it keeps dividing the chances.

The next row would be 6,25% –  25% – 37.5% – 25% – 6.25% and so on. If you add all 16 rows (16 pegs at the bottom row) the ball will have 2^16 possible paths it can go. That’s a total of 65,536 paths!

Exponential decrease in chance

Now, to calculate the probability of the ball ending up at the far left or far right, all we must do is divide the chance above with 2. All the way down to the bottom.  

Row              Chance in %

2                    50% / 2 = 25%

3                    25% / 2 = 12.5%

4                    12.5% / 2 = 6.25%

5                    6.25% / 2 = 3.125%

6                    3.125% / 2 = 1.5625%

7                    1.5625% / 2 = 0.7812%

8                    0.7812% / 2 = 0.3906%

9                    0.3906% / 2  = 0.1953%

10                  0.1953% / 2 = 0.0976%

11                  0.0976% / 2 = 0.0488%

12                  0.0488% /2 = 0.0244%

13                  0.0244% / 2 = 0.0122%

14                  0.0122% / 2 = 0.0061%

15                  0.0061% / 2 = 0.0030%

16                  0.0030% / 2 = 0.0015%

Because there’s a 0.0015% chance that the ball ends up either at the far left or the far right, there’s a total of 0.003% chance of the ball hitting the highest payout.

But what does 0.003% chance really mean? Statistically speaking it means that for every 33,333 ball you are dropping, 1 ball should (statistically) have hit the far right or the far left. But that’s just how it works in theory. In reality, you could drop 2 balls and have both hitting either far right or far left. That is however very unlikely. It is also possible to play 33,333 games without hitting far left or far right a single time. Because the probability, or chance, is the same for every ball you drop.

We humans tend to make up our own logic, such as “the more times I have played without hitting the highest multiplier, the higher the chance that my next game will hit the highest multiplier”.

That’s simply not true. Let’s take a coinflip as an example. There’s a 50/50 chance to hit either heads or tail when flipping a coin. Let’s say that we get heads on our first flip. On our second flip, we still have 50/50 chance to hit either heads or tail. The same goes for the third flip, the fourth and fifth and so on. The coin, physics and math doesn’t keep track on how many times you have flipped your coin, to adjust the odds for one or the other side to end face up. Every flip is a whole new flip, with the exact same odds. What you can do is to calculate what the odds are for getting i.e., 5 heads in a row. But I won’t cover that in this post, it will eventually get a post of its own.

Conclusion

Each time you play a game of Plinko, there’s a 1 in 33,333 chance that your ball will end up either at the far right or far left multiplier (at this example with 16 rows shows). This doesn’t mean that you can only hit the highest multiplier once every 33,333 games, it is just a measure of the probability of hitting the highest multiplier. You may very well hit the highest multiplier 2 times in 10 games, or no times in 50,000 games.

You can however check to make sure that the casino isn’t deceiving you, by checking the provable fair algorithm. And if you don’t trust the result on the provable fair link (provided for every in-house game), you can always calculate the result yourself.

Find out more about the probability of winning vs losing for various games: Why Is It So Hard To Win?

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