Determining the House Edge (Another math exercise)

Imagine a small jar of exactly 100 jelly beans, with each jelly bean colored either black or white.  Imagine a game where you pick a jelly bean from the jar at random, and pay $1 for the privilege.  If you select a white jelly bean you win a dollar!  (You get back $2, for a profit of $1.)  If you select a black jelly bean you lose.

Obviously, if there are exactly 50 jelly beans of each color in the jar you have an even gamble.  If this were a casino game the House Edge on this game would be zero.

If there are 51 black beans in the jar and only 49 white beans you would be at a 2% disadvantage.  That 2% would be considered the House Edge.  A jar containing 55 black beans and 45 white beans would mean you are up against a 10% House Edge.

I use this jar of jelly beans example because it helps to illustrate and visualize what exactly the House Edge represents.  If you bet on a proposition that has a 20% House Edge, your "jar" has 60 black beans and only 40 white beans!  As you can now see, this is not an ideal game to play!

Question:  What is the House Edge for each of the following five propositions:

(1)  A straight bet at -110. (An $11 bet wins $10 (returns $21) if it wins.)

(2)  A double-zero roulette wheel (38 slots) paying 35 to 1.

(3)  A 3-team parlay (No ML... each leg is at -110) paying 6 to 1.

(4)  A 4-team parlay (No ML... each leg is at -110) paying 12 to 1.

(5)  A 5-team parlay (No ML... each leg is at -110) paying 24 to 1.

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I will save you some time by providing the answers immediately.

There is a simple formula to determine the House Edge.  Actually, there are undoubtedly other ways to compute it - that's often the case with mathematics - but this is my favorite:

1 minus (total amount returned to you when you win divided by the total amount returned at fair odds, if there were no house edge)

(1)  Everyone interesting in NFL handicapping should immediately know the answer to this question, without having to calculate it.  It should be common knowledge.  No, the answer is not 5% nor is it 10%.  The House Edge on straight bets is just 4.545%.  This answer can be found on many websites such as https://wizardofodds.com/games/sports-betting/)

To arrive at the answer mathematically, just use the above formula.  The amount returned to you is $21.00 (your $11 bet and the $10 you won) divided by $22, the amount returned if were no House Edge.

1 - (21/22) = .045454 = 4.545%

When you make an 11 to win 10 straight bet, you are betting on a proposition that represents a jelly bean jar with 47.7 white beans and 52.3 black beans.  (This helps to show why you need to pick games at better than a 52.3% ratio to come out ahead.)

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(2)  Many readers will also know the House Edge on a double-zero roulette wheel off the top of their head.  It's 5.263%, a number that can also be verified at hundreds of other websites.  Let's use our formula to determine that answer.  As stated in the problem, a winning bet on a roulette wheel pays 35 to 1.  Including your $1 wager, you're paid back $36.  If there were no House Edge, you would have been paid at odds of 37 to 1 or $38 returned to you, once you add in the original dollar bet.  

1 - (36/38) = .05263 = 5.263%

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(3)  A 3-team parlay, with NO House Edge, would pay 7 to 1.  (1 chance to win out of 8 = 7 to 1 against you.)  A return of only 6 to 1 gives us a House Edge of

1 - (7/8) = .125 = 12.5%.  

This proposition using our jelly bean jar example give you approximately 43.75 white beans in the jar vs. 56.25 black beans.

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(4)  A 4-team parlay, with NO House Edge, would pay 15 to 1.  (1 chance out of 16 = 15 to 1 against you.  If the payout is only 12 to 1 this gives us a House Edge of

1 - (13/16) = .1875 = 18.75%.  

This jelly bean jar has approximately 40.6 white beans up against 59.4 black beans!  Suddenly a four-team parlay shouldn't look as attractive to you now, I'd like to hope!

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(5)  A 5-team parlay, with no House Edge, would pay 31 to 1.  (1 chance out of 32 = 31 to 1 against you.  A payout of 24 to 1 gives us a House Edge of

1 - (25/32) = .2187 = 21.87%.

Your jelly bean jar has just 39 white beans and 61 black beans!

When you play parlays, you are almost always paying a very high price to do so.  There are many more black jelly beans in your jar than white beans.  And yet the average bettor is blind to this.  All that see is that "big" return on their investment.

u wot m8

Math class at covers.  I love it.

Keep up the good work Ed .

excellent threadif only more people could take the time to a) do this kind of thing b) read this kind of thing

Next week, Ed, I want you to give us a lesson on the teaser. Lots of debate about the value of teasers on this forum (I'm no good at them). But I'm sure that's a lot tougher to break down mathematically. 

To sum it up,, if you have no edge on the game,, dont play,, games that are mathematically unbeatable such as table games in a casino are guaranteed profit for the house every year... 

Having said all that,, leagues spend millions upon millions every year recruiting,, hiring analyst,,,consultants,, and scouts,,... I suspect greater than 50% percent of them get it wrong,, Spending big money on players that cant last or just doesnt have what it takes ,, or thrown into a bad team from the get go... It is utterly a gift from above to measure talent on any human being with accuracy... More probabilities than and super computer nor man can measure...  

The problem with this is, picking jelly beans from a jar is 100% luck based proposition, capping games is far more skilled based.

Not sure what your point is here. It doesnt matter whether you get there by luck or skill or sorcery, if you win more than 52.4% of your bets at -110 odds, you're making money.

A 3 Team Parlay looks slightly more attractive now then. 

Nice thread Ed.

Ed, about 2 weeks ago I showed you the CORRECT math proving that the house edge is the same or less on a three team parlay than betting 3 flat plays. 
You ended up agreeing with me, remember?  Now you're saying  AGAIN that there's more house edge on the parlays, and its deceiving. 
You're forgetting to point out that flat plays cost 4.545% on EACH game so you have to add them together if you're betting more than one flat.  That makes the house edge difference better on the parlay, not worse.  And 6-1 is a better payout than if you pressed your money on three flat plays.  RIGHT?

Knowing math is one thing, drawing the right conclusions from the math is another....

Yes, Ed's initial math is correct, but then he skips the step showing how 6-1 is a better payout (LESS house edge) than 3 flat bets.  So he ends up with the wrong conclusion.

you can preach all these calculations but there will always be people that do 4 team parlays at 10 to 1 or 5 team parlays at 20 to 1 (off the board payouts), and then try to hedge the last legs and think they have just discovered the smart way to bet 

Hi Baby,

No, this is not correct.  If I agreed with you. I either agreed with you prematurely or we were talking about different things.

You don't add up individual House Edges and compare it to the House Edge of another single bet.  That's an invalid comparison.  It makes no sense.

Yes the House Edge on a single -110 bet is 4.54% and yes, 4.54 + 4.54 + 4.54 equals 13.62.  But you don't compare 13.62 to the House Edge on a 3-team parlay (12.5%) and say, "Oh, 12.5% is lower than 13.62%, so a three-team parlay has a lower house edge.  That makes no sense.

To help understand, assume I made 23 different bets, all with a House Edge of 4.54%.  If you add them together, that's 104.42.  A 104% House Edge?  What is that?  I don't even know what this is.  A guaranteed loss?  I don't think it's possible to have a House Edge that high. 

This should help to show that adding House Edges of individual bets makes no sense.  You never add them together for any reason.

So yes, the House Edge is more on a 3-team parlay than it is on three straight bets.  Each straight bet has a 4.54% house edge!  I don't know what wrong conclusion you think I'm drawing from this.

YES, if you ARE going to play a parlay card that pays 6 to 1, you ARE better off playing that parlay card, rather than parlaying each leg at -110 yourself.  (Assuming the games run at different times, otherwise you can't parlay it yourself anyway.)  The payout individually works out to 5.96 to 1 if I remember, which is slightly lower than 6 to 1.  So yes, the three-team parlay card at 6-1 is indeed better than parlaying the money yourself.  No dispute.  (I think this is the only parlay (a 3-teamer) where this is the case.)

Again, no.  I didn't skip a step.  There is no step to skip.  Again, this is an invalid comparison, as mentioned in my prior post.  This is something that is not done, (adding individual House Edges together, of individual events).

If I put a bet down on a roulette wheel, I had a 5.26% disadvantage.  If I then bet on a football game, at -110 bet, I had 4.54% disadvantage.  So together on the two bets I had a 9.8% disadvantage???

No.  You don't add these, ever.

Yes, I agree.  And if I can convince once person that's not ideal, and this costs him money in the long run, I'll be happy.

This is true.  But I didn't want Baby to think I wasn't aware of that a three-team parlay does pay more very slightly more than parlaying those three legs yourself.  But yes, you're correct... I never said anything regarding better payout.

If you're implying that a three team parlay looks slightly more attractive than a four team parlay, or a five team parlay, that's correct.

If you're implying that a three team parlay, paying 6 to 1, is more attractive than parlaying all three of those legs yourself, that's correct.

But if you're implying that a three-team parlay is more attractive than three individual bets, that's not correct.

My point is (one of them, anyway) that if you play parlay cards, you have to win at BETTER THAN 52.4% to make money!  Why?   Because the House Edge is higher!

After I showed you the math, you said "Yes of course."  We are/were talking about whether the house edge is worse on parlays rather than flat betting.
And yes, you DO keep adding in the house edge on the 3 flat plays, because you are PRESSING them;  you're never getting back even money so you're paying that house edge over and over again. 
Example...
110 only pays 100 so the house edge came into play, then I'm pressing the original 110 plus my 100 winnings onto the next flat bet.   210 only pays 191 so I'm paying the house edge again.  And so on and so on.  So I'm only getting a return of .909090 each time, which means I keep paying the house edge over and over against the same original money with each new flat bet.  That mimics a parlay, and comes out with almost no house edge difference. 
But as you stated I'd only get paid 5.96 on three flat plays, but the house is giving me 6.  So I'm actually getting a slightly BETTER return (LESS house edge) by playing the parlay rather than three flat bets.
Maybe I'm doing a lousy job explaining it, but see what I'm saying?  My whole point is the house is not screwing you with house edge on a 3 team parlay;  its the same or better than flat plays (assuming I try to press my luck).

But the house edge is DIRECTLY related to the payout.  That's how you're doing your calculations.  So better payout means lesser house edge....

ed