House Edge, Probability and RTP

Would you be able to work out the following house edge, probability and Return to Player etc for the following bet on a 23 number roulette wheel with a neighbour bet that covers three of the numbers. i.e a £100 wager would cover all three numbers on a roulette wheel with 23 pockets and play for up to three spins

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Neighbour bet - This wager will cover three numbers and is resolved across one, two or three spins. If the first spin matches any of the three wagered numbers, the player will win 5 times their wager, otherwise the wager is lost. The wager is not returned to the player if they win the first spin, instead it plays for a second spin.

If the second spin again matches the three wagered numbers then the player will win an additional 10 times their wager, otherwise the wager is lost. The wager is not returned to the player if they win the second spin, instead it plays for a third spin.

If the third spin again matches the the three wagered numbers then the player will win an additional 60 times their wager and the wager will be lost.

Many Thanks!

Quote: thomasandmiguel

Would you be able to work out the following house edge, probability and Return to Player etc for the following bet on a 23 number roulette wheel with a neighbour bet that covers three of the numbers. i.e a £100 wager would cover all three numbers on a roulette wheel with 23 pockets and play for up to three spins

.
Neighbour bet - This wager will cover three numbers and is resolved across one, two or three spins. If the first spin matches any of the three wagered numbers, the player will win 5 times their wager, otherwise the wager is lost. The wager is not returned to the player if they win the first spin, instead it plays for a second spin.

If the second spin again matches the three wagered numbers then the player will win an additional 10 times their wager, otherwise the wager is lost. The wager is not returned to the player if they win the second spin, instead it plays for a third spin.

If the third spin again matches the the three wagered numbers then the player will win an additional 60 times their wager and the wager will be lost.

Many Thanks!
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The first thing that we establish is that there are 23 numbers and the player is essentially betting on three of them at once. We're going to make the bet amount $5 for this.

I'm going to do the best I can with this, but I will say that I don't quite know what I get from your wording. For the second and third spins, when you say, "Additional," do you mean on top of the results from previous spins, or do you mean that is the total amount won?

Also, it seems that you lose your wager no matter what happens, so we start from a basis of -$5 and just have to go from there:

First Step Win and then Loss: (3/23) * (20/23) * 5 * 5 = 2.83553875236

First Step Win, Second Step Win, then Loss if 15x total: (3/23) * (3/23) * (20/23) * 5 * 15 = 1.10955864223

First Step Win, Second Step Win, then Loss if 10x total: (3/23) * (3/23) * (20/23) * 5 * 10 = 0.73970576148

Win/Win/Win if 75x total: (3/23)^3 * 75 * 5 = 0.83216898167

Win/Win/Win if 60x total: (3/23)^3 * 60 * 5 = 0.66573518533

SUM (If Wins Stack): 2.83553875236 + 1.10955864223 + .83216898167 = 4.77726637626

SUM (If Wins DO NOT Stack): 2.83553875236 + 0.73970576148 + 0.66573518533 = 4.24097969917

Expected Loss if Stack: 5 - 4.77726637626 = 0.22273362374

Expected Loss if Not Stack: 5 - 4.24097969917 = 0.75902030083

House Edge if Stack: .22273362374/5 = 0.04454672474 or 4.454672474%

House Edge if Not Stack: .75902030083/5 = 0.15180406016 or 15.180406016%

Again, this all assumes that the $5 wager is taken regardless of what happens with the spins, which is how you seemed to have phrased it. If you want to know it for the player retaining the wager upon three consecutive winning spins, then just replace '75' with '80' in the Win/Win/Win if stack and '60' with '65' if no stack and go from there. You'll also have to determine the probability of the player losing the $5 wager, which you can basically do the same way I did above.

Return to Player is just the inverse of the House Edge relative to one. If you want the RTP, just subtract the decimal form of the House Edges above from 1.

For the probabilities, again, your phrasing leads me to the impression that the original $5 bet is not to be returned no matter what. With that, here are the probabilities:

Spin One Loss: 20/23 = 0.86956521739

Spin One Win, then Loss = 3/23 * 20/23 = 0.11342155009

Spin One Win, then Spin Two Win, then Loss = 3/23 * 3/23 * 20/23 = 0.01479411522

All Three Win: (3/23)^3 = 0.00221911728

Please note I edited the above post to add the probabilities of each event.

P.s all odds would would be paid individually ie. First winning spin 5 for 1 (wager), 2nd winning spin 10 for 1 (with the same wager riding) and then 3rd spin 60 for 1 (the riding bet would then be lost)

I get the same numbers Mission146 gets - the house edge is 4.45467%

Probability of losing the bet on the first spin (result -1) = 20 / 23

Probability of winning on the first spin but losing the second (result 5 - 1 = 4) = 3 / 23 x 20 / 23 = 60 / 529

Probability of winning on the first two spins but losing the third (result 5 + 10 - 1 = 14) = 3 / 23 x 3 / 23 x 20 / 23 = 180 / 12,167

Probability of winning on all three spins (result 5 + 10 + 60 - 1 = 74) = 3 / 23 x 3 / 23 x 3 / 23 = 27 / 12,167