Bonus expected value in video poker

Going forward, expect me to start writing more about iGaming and less about gambling in land-based casinos. In this interest, I just played a bonus with these rules:

• 150% bonus given immediately.
• Play on video poker allowed.
• Best video poker game is 9-6 Jacks or Better (99.54% return).
• Max denomination is $5 ($25 total bet).
• 90x play requirement (deposit+bonus) to cash out.

The question I asked myself is what is my expected value, relative to the amount of the deposit? I was worried the 90x play requirement would kill any expected value. I deposited $500 for a starting bankroll of $1250. At $25 a bet I would have 50 units. At 90x play requirement, I would have to bet through 4,500 bets, or $112,500. On first glance it seemed like a huge task, where the house edge would surely grind me down?

However, most players can expect to bust out long before completing anywhere close to 90x. In reality, if you don't hit a royal early on, you probably won't last long. However, if you do, you should have plenty of ammunition to complete the play requirement.

However, I'm not big on sound bites and wanted some numbers, so I wrote a simulation.

The first table shows the probability of completing the play requirement for various starting bankrolls and play requirements. For example, the probability of completing the play requirement with a bankroll of bet 50 bets and 100x play requirement is 12.3%.

Initial Bankroll	20X	40X	60X	80X	100X	150X	200X
10	24.1%	15.4%	11.7%	9.5%	8.1%	5.9%	4.7%
20	32.1%	19.9%	14.7%	11.8%	9.8%	6.9%	5.4%
30	37.6%	23.0%	16.7%	13.1%	10.8%	7.6%	5.9%
40	41.9%	25.3%	18.1%	14.1%	11.6%	8.1%	6.4%
50	45.6%	27.2%	19.3%	15.0%	12.3%	8.6%	6.9%
60	48.7%	28.9%	20.3%	15.7%	12.8%	9.1%	7.5%
70	51.5%	30.3%	21.2%	16.3%	13.4%	9.6%	8.1%
80	53.9%	31.6%	22.1%	16.9%	13.9%	10.2%	8.6%
90	56.1%	32.8%	22.8%	17.5%	14.4%	10.8%	9.2%
100	58.2%	33.9%	23.5%	18.1%	15.0%	11.3%	9.8%

The next table shows the average bankroll when the player either busts out of completes the play requirement. For example, with an initial bankroll of 50 bets, and a 100x play requirement, the final outcome is a bankroll of 42.75 units.

Initial Bankroll	20X	40X	60X	80X	100X	150X	200X
10	9.58	9.40	9.28	9.18	9.10	8.94	8.84
20	18.97	18.49	18.19	17.96	17.76	17.37	17.12
30	28.25	27.49	26.93	26.53	26.23	25.59	25.15
40	37.50	36.36	35.53	34.94	34.58	33.61	32.98
50	46.76	45.14	44.07	43.24	42.75	41.74	40.75
60	55.90	53.86	52.57	51.54	50.78	49.30	48.43
70	65.08	62.56	60.92	59.86	59.00	56.96	55.66
80	74.19	71.14	69.38	67.93	66.84	64.67	63.11
90	83.22	79.80	77.58	75.99	74.74	72.41	70.20
100	92.48	88.32	85.86	84.10	82.66	79.31	77.14

The next table shows the average bankroll only when the player completes the play requirement. For example, with an initial bankroll of 50 bets, and a 100x play requirement, the final outcome, when the player is successful is a bankroll of 348.86 units.

Initial Bankroll	20X	40X	60X	80X	100X	150X	200X
20	59.17	92.78	123.42	152.82	181.21	250.43	316.03
30	75.12	119.61	161.42	201.88	242.17	337.27	425.05
40	89.41	143.73	195.96	247.11	297.70	414.14	515.15
50	102.58	165.93	227.99	289.15	348.86	483.44	588.10
60	114.77	186.68	258.50	328.81	396.06	540.41	645.53
70	126.44	206.15	286.96	366.36	440.50	591.86	691.35
80	137.67	225.05	314.61	401.16	481.42	632.17	729.85
90	148.25	243.21	340.72	433.51	517.62	671.77	759.11
100	158.97	260.51	365.70	465.46	552.58	701.53	788.30

In my case, I didn't run through a 90x play requirement exactly, but with my starting bankroll of 50 bets (or 250 coins), at an 80x play requirement the average outcome is 43.24 bets, and at 100x it is 42.75. Taking the average for 90x we get 43.00 units. Considering I actually purchased only 20 bets ($500/$25), this an expected profit of 115%.

I must say, that is more than I was expecting. It just goes to show that you shouldn't fear a huge play requirement if you can play a low house edge game and/or make large wagers.

Before I write about this on my Odds site, I welcome all comments and questions here.

Posts about new player registration at Lucky Red have been moved to Freaking out at Lucky Red.

90x play through. WHOLLY F

That is just crazy. Even after reading your page I wouldn't give it a shot. It is way to dependent on hitting a Royal.

4500 bets to cycle through 90x, wow.
And who doesn't make a mistake or multiple mistakes over 4500 bets on JOB.
The strategy isn't complicated but still, 4500, you know some mistakes will be made. What a grind.
I've put in 100 dollar bill in a quarter machine, that's 80 units and played it down to zero, no way I came close to 4500 bets. Boy that's real real tough.

Quote: Wizard

The question I asked myself is what is my expected value, relative to the amount of the deposit? I was worried the 90x play requirement would kill any expected value.

how worried were you?

Quote: Wizard

In my case, I didn't run through a 90x play requirement exactly, but with my starting bankroll of 50 bets (or 250 coins), at an 80x play requirement the average outcome is 43.24 bets, and at 100x it is 42.75. Taking the average for 90x we get 43.00 units. Considering I actually purchased only 20 bets ($500/$25), this an expected profit of 115%.

well i get confused fast when not seeing the math.
do you like my shoes? I was at Disneyland



yes,
How did you get to the return of 115%?

I show how I doooos it
VP for winners can show the probabilities but not the distribution


a free program I have does show the distribution, as well as my excel and is faster than VPW

for those 4500 bets I show an average loss of -7.01310923 units
0.00000001 (rounding error by program)

0.13460614 (survive) * 306.324886 (avg unit win given survived) = 41.23321049 +
0.86539386 (bust) * -20 (bankroll risked) = -17.3078772
ev = 23.92533329
is the return simply ev/20 or 1.196266665 * 100

Quote: Wizard

I must say, that is more than I was expecting.

for the record, how much were you expecting?

Sally

My take from this is now I understand why someone would spend $2.3M to promote Online Casino Bonuses. The Bonus sounds like a good deal until you do the math. Factor in that no one playing online can be 100% sure the Royals will show up in expected quanities and you have to have huge balls to try and get your expected "value" back.

But as always, Thanks to the Wizard for doing the math in a non-biased accurate way. Anyone who reads this and takes the chance cant say they didnt understand the odds.

Quote: Boz

The Bonus sounds like a good deal until you do the math.

I thought the Wizard's example showed a positive return to the player
of course, in my opinion, If you play this one time, there is no average return
unless you get that average by dividing by 1

in other words, do it many many many times to approach the calculated average (or simulated average)

of course, this assumes the online games are 100% honest, and there has never been shown any of them are 100% honest all of the time
so beware future readers

Quote: Boz

Factor in that no one playing online can be 100% sure the Royals will show up in expected quanities and you have to have huge balls to try and get your expected "value" back.

that expected value again was shown to be positive, was it not?

and why do you "have to have huge balls"
to try.
His buy-in was $500
Does the Wizard have huge balls?

what has that got to do with huge balls?

your statement is almost rude and offensive to me (that close)
were you trying to be "funny"

Quote: Boz

But as always, Thanks to the Wizard for doing the math in a non-biased accurate way. Anyone who reads this and takes the chance cant say they didnt understand the odds.

sure, thanks

+EV is what it is
Sally

The variance appears to be less than I expected (I would have guessed only 1 time in 25 you'd get to play through).

Obviously with the numbers and average results, you can work out an EV of your deposit, and a variance of the EV. If you can find enough at that rate, or better, you can repeat the trial over and over again. And then work out the value per hour. 4500 hands at 600 hands per hour is a days work... if you can get the play through that quickly.

Assume of course no malfeasance by the game operator, in terms of results or cash outs. I would suggest that being found it through LCB would be a fine service and information. Of course, payment for sign ups might narrow down for the LCB.com people if they were -too- awkward.

Quote: mustangsally

I thought the Wizard's example showed a positive return to the player
of course, in my opinion, If you play this one time, there is no average return
unless you get that average by dividing by 1

in other words, do it many many many times to approach the calculated average (or simulated average)

of course, this assumes the online games are 100% honest, and there has never been shown any of them are 100% honest all of the time
so beware future readers
that expected value again was shown to be positive, was it not?

and why do you "have to have huge balls"
to try.
His buy-in was $500
Does the Wizard have huge balls?

what has that got to do with huge balls?

your statement is almost rude and offensive to me (that close)
were you trying to be "funny"
sure, thanks

+EV is what it is
Sally

My point is that you have to assume the game is fair and you will get the expected number of Royals for his numbers to work. And you have to be very brave (Hows that for a non-offensive term?) to assume that with any online casino. I cant prove they are NOT honest, but its also hard to prove they are when opportunity knocks and there is little oversight.

+EV isnt always "what it is" when you cant be sure the game you are playing is what you are playing JMO.

The solution technique to this problem is not the correct approach. It has the same error that has existed in the Wizard's page on loss rebates that nobody has corrected for over a year. Hint: why didn't he talk about the strategy?

This is some excellent work by the Wiz.
It really gives an idea how tough it is to make a score with the bonus system.
I was really hoping our resident Bonus expert, Zuga , would chime in with some input.
Nothing, silence, weird.
This stuff is right up his alley.
I was hoping for a comment such as "Great Work Wiz, we need to bring this to the attention of our LCB members"
It seems Zuga pops in to oversee things but no real participation . I'm disappointed.
Seems he only responds to posts concerning LCB and that's it.
I would love for him to maybe start a thread about bonuses, the advantages and disadvantages. He is an expert.
I guess this is wishful thinking on my part.

Thank you for all the comments, especially Sally. I ran a simulation with exactly 50 units and 90x play requirement to compare my results to Sally's. They are as follows:

Probability survival:

Wizard: 13.45%
Sally: 13.46%

Expected loss (based on on starting point of 50 units):

Wizard: 7.1 units
Sally: 7.0 units

So, I'd say we're close enough.

You have small feet.

Quote: randomperson

The solution technique to this problem is not the correct approach. It has the same error that has existed in the Wizard's page on loss rebates that nobody has corrected for over a year. Hint: why didn't he talk about the strategy?

I assume you mean the player should play a more aggressive strategy. This is true. However, I didn't want to muddy the waters too much. There would be a different strategy for every different bonus situation. I wanted to show this could be done with no strategy changes.

To get back on topic, I just wrote a page about Bonus Expected Values in Video Poker. Please have a sneak preview. As always, I welcome all comments. You'll see there was actually a better game I should have been playing -- 8-5 Aces & Eights. Didn't know they had it when I originally posted. Shame on me.

Quote: Wizard

So, I'd say we're close enough.

yes, sure is nice when sim and calculated values are very close

Quote: Wizard

You have small feet.

size 6
ask your wife if you do not know
I bet she has small feet too

I am sure glad I do not have "big ass" feet (one of those is one too many)
I guess I stood in the correct line at the right time

small feet, hmmm
a female trait highly coveted by men
so many do say

another bonus
Sally

Quote: mustangsally

size 6
ask your wife if you do not know
I bet she has small feet too

I am sure glad I do not have "big ass" feet (one of those is one too many)
I guess I stood in the correct line at the right time

small feet, hmmm
a female trait highly coveted by men
so many do say

another bonus
Sally

I checked one of her shoes and it was a size 6.5. If I asked her I'd have had to go through a game of 20 questions about why I was asking.

Some of us men think that foot size in women is positively correlated to the size of ... another part of the body.

Quote: Wizard

I checked one of her shoes and it was a size 6.5. If I asked her I'd have had to go through a game of 20 questions about why I was asking.

Some of us men think that foot size in women is positively correlated to the size of ... another part of the body.

I believe that would be incorrect. For women, anyway. In my experience, it has also been incorrect for men.

We're talking noses, right?

Quote: Wizard

Some of us men think that foot size in women is positively correlated to the size of ... another part of the body.

????

Never heard that before. I always assumed girls just thought big feet on girls are ugly.

There's a thing about men with big feet.

Fun commercial dealing with that subject https://www.youtube.com/watch?v=Zafg3E8HIW8