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In this article, we’ll be breaking down the math behind the Plinko game on Stake.com. In Plinko, the player places a bet and then drops a ball down the board filled with pegs. The ball has a 50/50 chance at going to the left or right at each level. At the bottom of the Plinko board, there are buckets of payouts that are lowest in the middle and highest towards the edges of the board.Stake Casino: Play with Crypto and Win Big
The number of levels and the variance of the payouts can be changed in the game settings, resulting in a total of 27 different game options that are available to the player. We’ll be calculating the win probabilities and RTPs of each risk level to help you become more knowledgeable about the risks and rewards that this game offers.Stake Casino: Play with Crypto and Win Big
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Disclaimer: Stake may change their odds for the game at any point, meaning that the results of the calculations are not guaranteed to be exact at the time you’re reading this.
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Stake claims that Plinko has a 1.00% house edge, which is pretty player-friendly compared to odds offered by other online casinos and physical casino games. For reference, American roulette has a house edge of 5.26%.
In this section, we’ll be calculating the house edge ourselves to confirm that their stated house edge is correct.
Before we jump into the specifics of Plinko, let’s quickly go over how to calculate the expected value of a given game.
Let’s take a simplified example, where there’s a wheel that is split into 4 equally sized portions. It costs $10 to spin the wheel, and the 4 possible prizes are $3, $5, $12, and $20.
To calculate the expected value we can multiply each possible prize by the probability that we win that prize. Then we can add up all of those values. In this case, we’d add:
(25% x $2) + (25% x $5) + (25% x $12) + (25% x $20) = $9.75
So on average, we should win $9.75 from the wheel, but it costs $10 to spin, so the expected value of the full game would be:
$9.75 (Average Prize from Wheel) – $10 (Cost of Wheel Spin) = -$0.25 or -2.5%.
So now that we understand how to calculate the expected value of a game, let’s get into the specific probabilities of winning prizes in Stake’s Plinko game.
On Plinko’s Provably Fair page, they state that a ball should have a 50/50 chance of going left or right after falling through any given level.
The board below shows the payouts of 8 rows on the Low Risk mode. We can see that the payouts range from 0.5 to 5.6 times the original bet, with the smallest payouts in the middle and the largest payouts on the edges.
The buckets furthest from the middle are the highest because these are the most difficult bets to hit. The red path below shows how there’s only one way of making it to the far left bucket, and that’s by going to the left 8 times in a row (50% ^ 8 = 0.39% or 1 in 256 chance of occurring).
The buckets in the middle have the lowest payouts because they happen the most often. The blue and green show just 2 of the many possible paths that end up at the 0.5x multiplier.
So to calculate the expected value of Plinko, we’ll need to find the probabilities of the ball landing in each bucket, multiply by the prizes and add them all up.
To find the probabilities of the ball landing in a given bucket, we can use the binomial distribution shown below, where:
- n = The number of rows on the plinko board
- p = The probability of going left or right at a given row (50% in this case)
- x = The number of lefts needed to arrive at a given bucket
So, for example, if we wanted to calculate the probability of making it in the 5.6x bucket on the left side, we would have n=8, p=0.5, and x=8 (since we need to go left 8 times to make it to this bucket). After plugging this into the above formula, we get 0.39%.
To calculate the probability of making it in the 2.1x bucket on the left side, we would have n=8, p=.5, and x=8 (since we now only need 7 lefts and 1 right). This time, we get 3.125%, and we can see that the probabilities get higher as we get closer to the middle of the board.
Instead of calculating the above formula for each bucket, we can also plug the values into a Binomial Calculator like the one here.
After finding the probability of the ball landing in each bucket, we can then get the contribution to the expected value from each bucket, and after adding these up, we can see that the expected value of a $10 bet is -$0.102, or a 1.02% house edge, right around the 1.00% house edge that Stake listed.
| Bucket | 5.6 | 2.1 | 1.1 | 1 | 0.5 | 1 | 1.1 | 2.1 | 5.6 |
|---|---|---|---|---|---|---|---|---|---|
| Lefts Needed | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| Probability | 0.39% | 3.13% | 10.94% | 21.88% | 27.34% | 21.88% | 10.94% | 3.13% | 0.39% |
| Payout on $10 | $56 | $21 | $11 | $10 | $5 | $10 | $11 | $21 | $56 |
| Contribution to EV | $0.22 | $0.66 | $1.2 | $2.19 | $1.37 | $2.19 | $1.2 | $0.66 | $0.22 |
| Total Value of Game | $9.898 | ||||||||
| Total Expected Value Bet | -$0.102 | ||||||||
| Return to Player | 98.98% |
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We’ve calculated the Expected Value of 8 rows on Low Risk mode, but let’s test some other settings as well. In the below table we’ve calculated the house edge for each risk level with 8 and 16 rows.
We can see that all house edges ended up being right around 1.00%, but some are actually better than others. Out of the setting combinations we tested, the lowest house edge was 0.94%, and this came from 8 rows on the High Risk setting. The worst house edge was 1.09%, which came from 8 rows on the Medium Risk setting.
| Rows | Risk Level | House Edge |
|---|---|---|
| 8 | Low | 1.02 |
| 8 | Med | 1.09% |
| 8 | High | 0.94% |
| 16 | Low | 1.00% |
| 16 | Med | 1.01% |
| 16 | High | 1.02% |
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Overall, we’ve found that Stake takes about a 1% house edge for each setting combination of their Plinko game. However, we found that the house edges are not completely equal, and some settings actually have a lower house edge than others.
We’ve also covered the binomial distribution, and used this to calculate the expected value of a given Plinko game.
Thanks for reading, and I hope you’ve learned some useful information on calculating risks and rewards in games of chance!
