Gambling skill payouts

[Adam Dray]

The lottery game on page 131 is broken. It pays roughly 5.5 to 1 in favor of the ticket buyer. I wrote a little Java program to play 100 million games at a time and test out some scenarios.

-------------------- T5.09 Book Solution --------------------
You played 100000000 games.
You spent  1000000000Cr.
You won    1212974404Cr.

Result occurrences:
     Result:        0        1        2        3        4        5        6 
 Payout(Cr):        0        0       25      250     2500    25000    25000 
    #Occurs: 33482164 40189597 20099968  5358190   803828    64174     2079 

Your net earnings are 212974404Cr.
Your payoff ratio is 1.21 to 1.

-------------------- Adam's Solution --------------------
You played 100,000,000 games.
You spent  Cr100,000,000.
You won    Cr84,714,195.

Result occurrences:
     Result:        0        1        2        3        4        5        6        7        8 
 Payout(Cr):        0        0        0        0        5       15      200    20000   250000 
    #Occurs: 23239179 37228448 26047947 10422238  2603063   415632    41172     2274       47 

Your net loss is Cr15,285,805.
Your loss ratio is 1 to 1.18.

With a little trial and error, I think you want to do this instead:

To play the lottery (once per week)
Beyond Impossible (8D)

Pay Cr10 for a ticket.
Roll 8 ones and win Cr250,000.
Roll 7 ones and win Cr20,000.
Roll 6 ones and win Cr200.
Roll 5 ones and win Cr15.
Roll 4 ones and win Cr5.
Roll less than 4 ones and lose your money.​

Even those tend to pay close to parity, but there's a decent chance that, out of 100 million tickets sold, fewer will hit the higher payouts. 100 million tickets is probably in the neighborhood of a affluent, mid-sized planet's sales for one week.

Note that even small changes to the mid-range payouts dramatically changes the results. Note that, the more dice you add, the more you dramatically increase the chance of getting, say, 5 ones.

 

[Daddicus]

You are correct, but using a computer model doesn't work for accurately predicting the odds. Math is needed for that. Fortunately, I have some worthless piece of paper somewhere that says I'm supposed to know about such things.

They way to figure it out is to enumerate all possible dice combinations, and then assume each one comes up exactly once. So, for 2 dice, 2 6's come up one time in 36. A single 6 comes up 10 more times.

For 6 dice, there are 46,656 possible combinations. The house collects 466,560cr.

Of those, there is 1 combination that contains all 6's. Payout is once at the 250k prize level. So far, so good. The house is still ahead.

There are 30 ways to get 5 6's on 6 dice. These 30 get paid 25,000cr each, for a total of 750,000cr paid out.

The house just went bankrupt, and we haven't even looked at 4, 3, or 2 6's. It turns out the 4-sixes payout is the one that costs the house the most.

The house needs to either charge 60cr per ticket (which is darned close to break-even, slightly favoring the house). Or, they can raise the number of dice.

Personally, I'll use Chuck-a-Luck in my campaign. Put your money on a number, then roll 3 dice. If your number comes up, you get your money back, plus however many times your number showed up. It's 50/50, so payouts should be even, right? (No, but have fun figuring out why.)

 

[CosmicGamer]

In the first post, without seeing any code, something seams off

The Book Solution net earnings seam to be what you spent + what you won.
If you do the same with the Adam Solution don't you get a net earnings of Cr184,714,195 instead of a net loss?

In the second post, I'm confused as to how there are 30 ways to get 5 6's on 6 dice. I may be no expert in odds, but I think it is 6 ways.

 

[Daddicus]

In the second post, I'm confused as to how there are 30 ways to get 5 6's on 6 dice. I may be no expert in odds, but I think it is 6 ways.

I got tripped by that, too, before I built the complete list and counted them. Here's why: each of the 6 positions can be filled with a non-6. There are 5 non-6 results that can go into each of these slots, so it's 6 times 5.

666661
666662
666663
666664
666665

666616
666626
666636
666646
666656

666166

etc.

 

[Adam Dray]

In the first post, without seeing any code, something seams off

The Book Solution net earnings seam to be what you spent + what you won.
If you do the same with the Adam Solution don't you get a net earnings of Cr184,714,195 instead of a net loss?

In the second post, I'm confused as to how there are 30 ways to get 5 6's on 6 dice. I may be no expert in odds, but I think it is 6 ways.

Let me check my code and get back to you. :coffeesip:

 

[CosmicGamer]

There are 5 non-6 results that can go into each of these slots, so it's 6 times 5.

Thanks for the clarification.