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Ask the Wizard #93

I saw a bet in craps called the "Fire Bet" that paid if player made 4 to 6 different points. What is the scoop on that?

"Anonymous" .

The following table shows what each number of points pays, the probability, and contribution to the total return. The probabilities were determined by random simulation. The exact probability of making all six points is 0.000162.

Fire Bet

Points Made




0 0.594522 -1 -0.594522
1 0.260503 -1 -0.260503
2 0.101038 -1 -0.101038
3 0.033364 -1 -0.033364
4 0.008776 10 0.087764
5 0.001633 200 0.326582
6 0.000164 2000 0.328063
Total 1 -0.247017

The lower right cell shows an expected loss, or house edge, of 24.70%. It is my understanding the only allowed bet amount is $2.50, so the expected loss per bet would be about 62 cents.

What would be the probability of hitting a royal flush in video poker if you always played the best strategy to do so, which would consist of always keeping 1 or more to a royal and throwing away all cards that don’t compose a royal flush? What would be the housed advantage in this situation? Just curious. Thanks.

"Anonymous" .

If your strategy were to maximize the number of royals at all costs then you would hit a royal once every 23081 hands. I assumed that given two plays of equal royal probability the player will choose the play which maximizes the return on the other hands. The house edge of this strategy on a 9/6 jacks or better game is 51.98%. Below is a table showing the probability and return of each hand.

Royal Seeker Return Table

Hand Payoff Probability Return
Royal Flush 800 0.000043 0.034661
Straight Flush 50 0.000029 0.001472
4 Of A Kind 25 0.000222 0.005561
Full House 9 0.001363 0.012268
Flush 6 0.00428 0.025681
Straight 4 0.004548 0.018191
3 Of A Kind 3 0.020353 0.061058
Two Pair 2 0.046374 0.092749
Jacks Or Better 1 0.228543 0.228543
Nothing 0 0.694243 0
Total 0 1 0.480184

At Pinnacle Sports there is a "Multi-Way calculator on the right" that shows the house edge on money line bets. What is the formula they are using?

"Anonymous" .

This is interesting. Normally the house edge is lower betting on the favorite, as I explain in my sports betting appendix 3. However at Pinnacle they evidently set the money lines so that each has the same house edge. Let d be the money line on the dog and f be the money line on the favorite. For example if the money lines were +130 and -150 then d=130 and f=-150. The house edge on both bets at Pinnacle would be:


The amount you must bet to get back one unit is 1/[(d/100))*(1-(100/f))/(2+(d/100)-(100/f))].

For example with money lines of +130 and -150 the house edge on both bets would be 3.3613% and the expected return on a bet of 1.034783 units would be 1 unit.

At a land casino, I would assume the fair set of money lines to be +140 and -140 in this example, resulting in a house edge of 2.78% on the favorite and 4.17% on the dog. All other things being equal this would suggest that Pinnacle is a good place to bet on underdogs.

I have a coupon from LVA #115 Free Blackjack Insurance up to $25.00 at Slots of Fun. What’s its value?

"Anonymous" .

I have that coupon too, and am running out of time to use it. Let’s assume a single deck game. The probability the dealer has blackjack with an ace showing is 16/51 = 31.37%. So if you bet $50 the value of this coupon is (16/51)*$50 = $14.71. However I estimate you will lose $1.23 due to the house edge waiting for the opportunity to use it. So the coupon itself is worth $14.71 - $1.23 = $13.48.

Can you tell me about the legal penalties for cheating at a casino? For instance, could the casino press charges if you are caught using a mechanical card counting device, or just kick you out? How about other scenarios, like if you were caught with a computer system to predict roulette spins, or a device to spy on poker hands?

"Anonymous" .

It is my understanding that cheating in a Nevada casino carries the same penalty as bank robbery. Computers and cameras definitely count as cheating devices.

If I determine the fair line of a game to be -160/+160 and I find a rogue line of -145 what is my EV? Any formula you could provide in which I could derive my EV +/- after a fair line has been determined would be greatly appreciated.

"Anonymous" .

Let p be the probability of the favorite winning. If -160 is a fair line then:

100*p - 160*(1-p) = 0
260p = 160
p = 160/260 = 8/13 = 61.54%.

So the expected return on a $145 bet at a -145 line would be (8/13)*100 + (5/13)*-145 = 75/13 = $5.77. So the player advantage would be $5.77/$145 = 3.98%.

Let’s define t as the true money line with no house edge and a as the actual money line. Following are the formulas for the player’s expected return:

A is negative, t is negative: (100*(t-a) / (a*(100-t))
A is positive, t is positive: (a-t)/(100+t)
A is positive, t is negative: (a*t + 10000)/((t-100)*100)

So in your case your expected return is 100*(-160 -(-145))/(-145*(100-(-160))) = 3.98%.