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The Probability Of Winning At Plinko


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The origin of Plinko

The game of Plinko has its origin sometime in the late 1800s and was originally constructed by mathematician and statistician Francis Galton. He built what is called a "Galton box" to prove that with a large enough sample, a binominal distribution (i.e., when there are two options with an 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. 



Your balls would pretty much end up like this


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 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 one 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 two pegs, 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 left again, we must take the whole board into 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, we must 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 mean? Statistically, it means that for every 33,333 balls you drop, one ball should (statistically) have hit the far right or far left. But that's just how it works in theory. In reality, you could drop two balls and have both hitting either far right or far left. That is, however, very unlikely. Playing 33,333 games without hitting far left or far right a single time is also possible. 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 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 a 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 don't keep track of 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. You can calculate the odds of getting, i.e., five heads in a row. But I won't cover that in this post; it will eventually get a post of its own. 


Provable fair

Plinko is a provable fair game at BC.game. That means that the result is known way before the ball hits bottom. This is calculated through the client seed + nonce, and the server seed goes through an algorithm, which will give the game's result. Since the casino provides its seed (encrypted) before you start the game, the casino can't change it without it showing. 


Client seed: This is the seed you can change yourself.

Server seed: This seed is provided by BC.game.

Nonce: The number of times you have played using the client seed.





The hexadecimal numbers resulting from client seed, nonce and server seed are then converted to base10 numbers. 





These will, in turn, be calculated in groups of 4 to a number between 0 and 1. If the result from the first 4 base10 numbers is less than 0.5, the ball will go left; if it's higher than 0.5, it will go right.

The math for these four first numbers will look like this.




This will be the same as:




And equals the number: 0.6657153440173715.

So, it’s more than 0.5, which means the ball went right.

The next would be (100/256^1) + (82/256^2) + (79/256^3) + (240/256^4) = 0.39188098534941673 – This is less than 0.5, so the ball went left.




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 (as 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 hit the highest multiplier two times in 10 games or no times in 50,000 games.

You can, however, check to ensure 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.




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This was very interesting to read, as somebody who is new to BC but has played a lot of Plinko. Sometimes going without the highest multiplier for double the theoretical odds. But also, more than once I've landed them in the highest multiplier on the 3rd try. 

Thanks for taking the time to write this. 

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43 minutes ago, Vorseline said:

This was very interesting to read, as somebody who is new to BC but has played a lot of Plinko. Sometimes going without the highest multiplier for double the theoretical odds. But also, more than once I've landed them in the highest multiplier on the 3rd try. 

Thanks for taking the time to write this. 

I'm glad you liked it! Probabilities don't work the way most people think it does.




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