https://www.virgingames.com/?game-info=play-nice-dice

https://www.virgingames.com/game/play-nice-dice/demo

There was already an attempt to find the best strategy, by a Japanese guy here.

https://www.a-taichi.com/tablegame/nicedice.html

I was wondering if the community is up for the challenge.

GAME INFORMATION

The main aim of this game is to roll a scoring hand after three rolls of five dice.

You will have three rolls, with two rounds of holding your dice in-between.

You can choose to hold or not hold as many or as few of the five dice as you wish between rolls.

After the third roll, the game automatically resolves the final configuration of the five dice.

RESULT DESCRIPTION PAYS

Full House *** Three dice of the same value with two other dice of the same value *** Push

Straight *** Five consecutive valued dice *** 1:1

Four of a Kind *** Four dice of the same value and one other non-matching dice *** 3:2

Five of a Kind *** Five dice of the same value *** 3:1

Nice Dice *** Five of a kind on the first roll *** 9:1

Quote:MattUK(snip)The RTP is 99.54%, so I strongly believe they tried to make an alternative to Craps. (snip)

Maybe, but it seems closer to a "Yahtzee-like" game to me.

Also, in your post there is a description/copy paste error where it says, "Straight *** Three dice of the same value with two other dice of the same value *** 1:1" as it should be describing a "straight" (e.g. 1-2-3-4-5) but it is just repeating the definition of a "full house"

Here are some links that may be helpful/interesting:

https://en.wikipedia.org/wiki/Yahtzee

https://math.stackexchange.com/questions/320889/yahtzee-large-straight-strategy

http://datagenetics.com/blog/january42012/index.html#:~:text=Finally%2C%20there%20is%20a%2010,or%20the%20second%20one%20does).

---

Spelling not checked thoroughly

Playing it for a while on demo helps to feel the best choices intuitively so I would recommend it. The link is in the first post.

Quote:MattUKThere was already an attempt to find the best strategy, by a Japanese guy here.

https://www.a-taichi.com/tablegame/nicedice.html

I was wondering if the community is up for the challenge.

Sounds easy enough, but I can't find the demo, so I have one question: if I hold (i.e. not reroll) a die for the second roll, can I then choose to reroll it in the third roll?

There is now a link to the demo version in the first post.

I suspect that - much like always going to war with ties in War - player should always try to create Five of a kind. The optimal strategy table may look like this, but this is probably an oversimplification.

Initial roll | First decision | Second decision |
---|---|---|

any instant winning hand except Four of a kind | stand | --- |

Four of a kind | roll not matching die | roll not matching die or stand if matched |

1-3-4-5-6 or 1-2-3-4-6 | Roll 1 or 6 | Roll 1 or 6 if not matched, otherwise stand |

2-3-4-5 and one dublet | Roll doubled die | Roll doubled die if still not Straight, otherwise stand |

Three of a kind | Roll not matching 2 dice | Roll not matching 2 or 1 dice, stand with Five of a kind |

One pair | Roll not matching 3 dice | Roll not matching 3, 2 or 1 dice, stand with Five of a kind |

Don't break up a full house, three of a kind, or two pair

Keep four to a straight even if it means breaking up a pair; otherwise, hold onto the pair

(i.e. hold onto a pair if the roll includes a 1 and a 6)

As ax example, suppose that, after your second roll, you have 1 1 2 2 6

If you hold the two pair, you have 1/3 chance of a full house (EV 0) and 2/3 of a loss (EV -1), for a total EV of -2/3

If you hold, say, the 1s, then, of the 216 rolls:

1 is 1,1,1 (EV +3)

15 are 1,1,x (EV +1.5)

15 are 1,x,x (EV 0)

5 are a different three of a kind (EV 0)

The other 190 are losing rolls (EV -1)

The total EV is (3 + 22.5 - 190) / 216 = about -0.7616

Play as seems obvious except

1st Round

- Full House, only keep Trips.

- Two pairs, keep one pair (K Q J or T if possible) - If you only keep the highest pair it's 99.68%.

- Pair, play the pair unless there's an easy straight draw (e.g. KKQJT).

- Nothing, go for the straight.

2nd Round

- Stand on Full House (it's a shame they've not quite made this an optional play where the EV is the same whether you stand or draw)

- Go for any straight draw (i.e. discard from the pair)

- Draw one card to Two Pairs.

Note: Sorry I didn't make it clear that I was using Poker Dice (easier to think), so for K Q J T read 5 4 3 2 and KKQJT read 55432. With Two Pairs on the 1st round, if you have a choice throw away the 11 or 66.

Quote:ThatDonGuyThe strategy after the second roll is:

You can't talk about the second decision without resolving the first one (what to do in all cases). Probably that's why you have that funny 23.84% RTP. It's like American Roulette with 29 zeros. Please consider looking on that Japanese website and translate using Google.

https://www.a-taichi.com/tablegame/nicedice.html

It's not easy to go through his tables, but he did it Wizard-style. Ideally, we should also have EV for any case to be able to compare them and see the increase (unless it was a risk worth taking, but the roll decreases the EV, which in principle may happen).

Here is his summary. I am trying to figure out what the oriental names describe.

https://drive.google.com/file/d/1lk10B2ZT5NfidfwgKf-ZDxDXPlTEGZdS/view?usp=sharing

Quote:charliepatrick

- Two pairs, keep one pair (K Q J or T if possible) - If you only keep the highest pair it's 99.68%.

- Pair, play the pair unless there's an easy straight draw (e.g. KKQJT).

- Draw one card to Two Pairs.

Ehm... Charlie... it's a wrong thread. This one is about a dice game. :-)

Possibly you missed my earier comment that I was going to use poker dice as it makes it easier to think about Straights etc. So it was easier for me to see that "KQJT" is an outside straight draw than "5432". Sorry if that caused confusion.Quote:MattUKQuote:charliepatrick

- Two pairs, keep one pair (K Q J or T if possible) - If you only keep the highest pair it's 99.68%.

- Pair, play the pair unless there's an easy straight draw (e.g. KKQJT).

- Draw one card to Two Pairs.

Ehm... Charlie... it's a wrong thread. This one is about a dice game. :-)

Quote:MattUKYou can't talk about the second decision without resolving the first one (what to do in all cases). Probably that's why you have that funny 23.84% RTP. It's like American Roulette with 29 zeros. Please consider looking on that Japanese website and translate using Google.

https://www.a-taichi.com/tablegame/nicedice.html

It's not easy to go through his tables, but he did it Wizard-style. Ideally, we should also have EV for any case to be able to compare them and see the increase (unless it was a risk worth taking, but the roll decreases the EV, which in principle may happen).

Here is his summary. I am trying to figure out what the oriental names describe.

https://drive.google.com/file/d/1lk10B2ZT5NfidfwgKf-ZDxDXPlTEGZdS/view?usp=sharing

Here's the first roll strategy:

Five Of A Kind - hold all five

Four Of A Kind - hold the four; reroll the other die

Straight - hold all five

Full House - hold all five

Three Of A Kind - hold the three; reroll the other two dice

Two Pair - hold one of the pair; reroll the other three dice

2345 and a Pair - keep 2345, and break up the pair

One Pair (other than with 2345) - hold the pair, and reroll the other three

12346, 12356, 12456, 13456 - reroll the 1 and the 6, and keep the other three

The RTP is 99.5376324%

Here's the complete strategy:

Five Of A Kind - hold all five

Four Of A Kind - hold the four; reroll the other die

Straight - hold all five

Full House - hold all five

Three Of A Kind - hold the three; reroll the other two dice

Two Pair

First roll: hold one of the pair; reroll the other three dice

Second roll: hold both pairs; reroll the other die

One Pair

First roll: if you have 2345, keep the 2345, and break up the pair; otherwise, hold the pair, and reroll the other three

Second roll: if you have a 1 and a 6, hold the pair, and reroll the other three; otherwise, keep the 4 to a straight (even an inside straight), and break up the pair

12346, 12356, 12456, 13456

First roll: reroll the 1 and the 6, and keep the other three

Second roll: reroll the 1 (or the 6), and keep the other four

Initial roll | First decision | Second decision |
---|---|---|

Five of a kind | stand | stand |

Four of a kind | reroll unmatched die | reroll unmatched die |

Straight | stand | stand |

Full House | hold Three of a kind, reroll the other two dice | stand |

Three of a kind | reroll the other two dice | reroll the other two dice |

Two Pair | hold inside Pair if you can; reroll the other three dice | hold both pairs; reroll the other die |

One Pair and 2345 | hold the 2345; reroll the paired die | hold the 2345; reroll the paired die |

One Pair and one Outsider* | hold the pair, and reroll the other three | hold 4 to a Straight, and reroll the paired die |

One Pair and both Outsiders* | hold the pair, and reroll the other three | hold the pair, and reroll the other three |

13456, 12456, 12356, 12346 | reroll both Outsiders*, and hold the other three | reroll either Outsider*, and hold the other four |

* Outsiders are numbers 1 and 6

The above table takes into account subsequent discussion and corrections.

Initial roll | First decision | Second decision |
---|---|---|

One Pair and 2345 | hold the 2345; reroll the paired die | hold the 2345; reroll the paired die |

One Pair, and either but not both 1 or 6** | hold the pair, and reroll the other three | hold 4 to a straight, and reroll the paired die |

One Pair, and both 1 and 6** | hold the pair, and reroll the other three | hold the pair, and reroll the other three |

** note the 1 and 6 may or may not be in the pair; for example, both 1 2 4 4 5 and 2 3 5 6 6 are "one pair, and either but not both 1 or 6," and both 1 1 2 3 6 and 1 4 4 5 6 are "one pair, and both 1 and 6"

Initial Roll Outcome | ||
---|---|---|

Nice Dice (AAAAA) | 6 | 10.000 |

Quads (AAAAK) | 150 | 2.958 |

Straight (AKQJT) | 240 | 2.000 |

Trips or FH (AAA??) | 1 500 | 1.539 |

Easy Str Draw (KKQJT) | 240 | 1.111 |

2 Pair As & 9s (AA99x) | 120 | 0.742 |

2 Pairs (AAKKx) | 1 680 | 0.755 |

Pair As or 9s (AAxxx) | 1 200 | 0.742 |

Pair Ks Qs Js Ts (KKxxx) | 2 160 | 0.755 |

Inside Straight (AKQJ9) | 480 | 0.634 |

First Roll | ||

Nice Dice (AAAAA) | 10.000 | Stand |

Fiver (AAAAA) | 4.000 | Stand |

Quads (AAAAK) | 2.958 | Draw one |

Full House (AAAKK) | 1.000 | (i) Stand |

Full House (AAAKK) | 1.539 | (ii) Keep Trips |

Straight (AKQJT) | 2.000 | Stand |

Trips (AAAxx) | 1.539 | Keep Trips |

Pair / E Str (KKQJ T) | 0.755 | (i) Keep Pair (K Q J or Ts) |

Pair / E Str (KKQJ T) | 1.111 | (ii) Straight Draw |

Pair / I Str (AAKQJ) | 0.742 | (i) Keep Pair (As or 9s) |

Pair / I Str (AAKQJ) | 0.611 | (ii) Straight Draw |

Pair / I Str (KKxxx) | 0.755 | (i) Keep Pair (K Q J or Ts) |

Pair / I Str (KKAQJ) | 0.611 | (ii) Straight Draw |

Two Pair (AA99x) | 0.742 | (i) Keep Pair (As or 9s) |

Two Pair (AA99x) | 0.556 | (ii) Keep Two Pair |

Two Pair (KK99x) | 0.755 | (i) Keep Pair (K Q J or Ts) |

Two Pair (KK99x) | 0.556 | (ii) Keep Two Pair |

Pair (As or 9s) (AAxxx) | 0.742 | (i) Keep Pair (As or 9s) |

Pair (Ks Qs Js Ts) (KKxxx) | 0.755 | (i) Keep Pair (K Q J or Ts) |

Inside Str (AKQJ9) | 0.611 | (i) Keep 4 to Str |

Inside Str (AKQJ9) | 0.634 | (ii) Keep KQJ |

Inside Str (AKQT9) | 0.611 | (i) Keep 4 to Str |

Inside Str (AKQT9) | 0.634 | (ii) Keep KQT |

Inside Str (AKJT9) | 0.611 | (i) Keep 4 to Str |

Inside Str (AKJT9) | 0.634 | (ii) Keep KJT |

Inside Str (AQJT9) | 0.611 | (i) Keep 4 to Str |

Inside Str (AQJT9) | 0.634 | (ii) Keep QJT |

Second Roll | ||

Nice Dice (AAAAA) | 10.000 | Stand |

Fiver (AAAAA) | 4.000 | Stand |

Quads (AAAAK) | 2.750 | Draw one |

Full House (AAAKK) | 1.000 | (i) Stand |

Full House (AAAKK) | 0.944 | (ii) Keep Trips |

Straight (AKQJT) | 2.000 | Stand |

Trips (AAAxx) | 0.944 | Keep Trips |

Pair / E Str (KKQJ T) | 0.285 | (i) Keep Pair |

Pair / E Str (KKQJ T) | 0.667 | (ii) Straight Draw |

Pair / I Str (AAKQJ) | 0.285 | (i) Keep Pair |

Pair / I Str (AAKQJ) | 0.333 | (ii) Straight Draw |

Two Pair (AAKKx) | 0.285 | (i) Keep Pair |

Two Pair (AAKKx) | 0.333 | (ii) Keep Two Pair |

Pair (AAxxx) | 0.285 | Keep Pair |

Inside Str (AKQJ9) | 0.333 | Keep 4 to Str |

Return to player | 99.537 632% |

Quote:charliepatrickI now agree with the RTP but there's still the need to hold an inside Pair when you choose which Pair to hold from Two Pairs after the first roll. Also note you only keep the Trips after the first roll with a Full House, but you keep the Full House after the second roll.

Great! Now we're in agreement with the Japanese guy! He had 10 different outcomes, wrote "Hold one pair in the middle" for the first decision having Two Pair and that it's better to reroll two unmatched dice having Full House.

I am beginning to understand what ThatDonGuy had in mind. Somehow Mr. Toyoda did the same, but that was too fragmented piece of the puzzle to understand without background, sorry.

I corrected the table and added "if you can" because it's still possible to roll 1166 and inside number.

PS: Nice Dice is not formed after a decision. They can be removed from the bottom two return tables. Hope that doesn't change anything.

"First decision" table

Initial roll | First decision | Expected value |
---|---|---|

Five of a kind | stand | 4.000 |

Four of a kind | reroll unmatched die | 2.958 |

Straight | stand | 2.000 |

Three of a kind | hold three of a kind, reroll the other two dice | 1.539 |

Full House | hold three of a kind, reroll the other two dice | 1.539 |

insider Pair and 2345 | hold the 2345, reroll the paired die | 1.111 |

insider Pair and 1 or 6 | hold the pair, reroll the other three | 0.755 |

11 or 66 Pair | hold the pair, reroll the other three | 0.742 |

Two Pair 11 and 66 | hold either pair, reroll the other three | 0.742 |

any other Two Pair | hold insider pair, reroll the other three | 0.755 |

13456, 12456, 12356, 12346 | reroll 1 and 6, hold three insiders | 0.634 |

"Second decision" table

Second roll | Second decision | Expected value |
---|---|---|

Five of a kind | stand | 4.000 |

Four of a kind | reroll unmatched die | 2.750 |

Straight | stand | 2.000 |

Three of a kind | reroll the other two dice | 0.944 |

Full House | stand | 1.000 |

insider Pair and 2345 | hold the 2345, reroll the paired die | 0.667 |

insider Pair and 1 or 6 | hold 4 to a Straight, reroll the paired die | 0.333 |

Two Pair 11 and 66 | hold both pairs, reroll the other die | 0.333 |

any other Two Pair | hold both pairs, reroll the other die | 0.333 |

13456, 12456, 12356, 12346 | reroll 1 or 6, hold the other four | 0.333 |

11 or 66 Pair | hold the pair, reroll the other three | 0.285 |

Strategy for Two Pairs can be merged into "hold insider Pair if you have it, otherwise either pair; reroll the other three". Likewise, some cells can be merged if EV is ditched.

Note that the EV decrease in the second round for every outcome, except instant winners Five of a kind and Straight.

For the avoidance of doubt, Yahtzee aka Nice Dice returning 10x is an automatic winner after the initial roll and hence cannot be formed.

Just maybe some people would like to have all five columns in a landscape and print out horizontally. It can be done, but not exactly using html.

And finally, here's their "handy video tutorial":

https://www.linkedin.com/posts/roxor-gaming_roxorgaming-wemakeplaymorerewarding-igaming-activity-6689475826964033536-aZk9/

Quote:MattUKIt was laborious, but here is the final answer. My thanks to charliepatrick and thatdonguy for doing the calculations.

"First decision" table

Initial roll First decision Expected value Five of a kind stand 4.000 Four of a kind reroll unmatched die 2.958 Straight stand 2.000 Three of a kind reroll the other two dice 1.539 Full House hold Three of a kind, reroll the other two dice 1.539 insider Pair and 2345 hold the 2345, reroll the paired die 1.111 insider Pair and 1 or 6 hold the pair, reroll the other three 0.755 11 or 66 Pair hold the pair, reroll the other three 0.742 Two Pair 11 and 66 hold either pair, reroll the other three 0.742 any other Two Pair hold insider pair, reroll the other three 0.755 13456, 12456, 12356, 12346 reroll 1 and 6, hold three insiders 0.634

"Second decision" table

Second roll Second decision Expected value Five of a kind stand 4.000 Four of a kind reroll unmatched die 2.750 Straight stand 2.000 Three of a kind reroll the other two dice 0.944 Full House stand 1.000 insider Pair and 2345 hold the 2345, reroll the paired die 0.667 insider Pair and 1 or 6 hold 4 to a Straight, reroll the paired die 0.333 Two Pair 11 and 66 hold both pairs, reroll the other die 0.333 Tany other Two Pair hhold both pairs, reroll the other die 0.333 13456, 12456, 12356, 12346 reroll 1 or 6, hold the other four 0.333 11 or 66 Pair hold the pair, reroll the other three 0.285

The tables can be slightly simplified by giving recommendation "hold insider Pair if you have it, reroll the other three".

Note that the EV decreas in the second round for every outcome, except Five of a kind and Straight.

Nice table. Only commented is EV on first roll for five of a kind. I thought this got the nice dice bonus?

Quote:unJonNice table. Only commented is EV on first roll for five of a kind. I thought this got the nice dice bonus?

I skipped that because it does not require a decision, but it's included in the 99.54% RTP.

Questions:

1. Can I re-roll a die already placed in the tray?

2. If I'm happy with what I have after the first or second roll, must I re-roll? If not, how does the player indicate he doesn't want any remaining rolls?

I would appreciate a nice screen-shot of the game from anybody who has password access.

Quote:Wizard1. Can I re-roll a die already placed in the tray?

2. If I'm happy with what I have after the first or second roll, must I re-roll? If not, how does the player indicate he doesn't want any remaining rolls?

I would appreciate a nice screen-shot of the game from anybody who has password access.

1. The tray is a nice representation of held dice. You can change your mind and un-click them before you press "ROLL".

2. You always can stand - by pressing the "STAND" button. If it's throwing away the stake you will be asked for confirmation as safety net.

3. as per your request - https://drive.google.com/file/d/1JuyoKI8ey54ZQn3zOfNR0Sm1j6ZkZIU7/view?usp=sharing

https://drive.google.com/file/d/1tQuGA70dPMElB_UH3pAGr9fHayfYY59z/view?usp=sharing

Nice job, but I wish there was also a chart showing the possibility of each hand at the end of 3 rolls. Kind of important when figuring the house edge.

Quote:DJTeddyBearMattUK -

Nice job, but I wish there was also a chart showing the possibility of each hand at the end of 3 rolls. Kind of important when figuring the house edge.

Sorry to disappoint. :-) The house edge is what it is - 0.46% when making best choices. Just between Pass Line and Don't Pass Line in Playtech Craps. A pity they did not best Don't Pass Line, in my opinion. This game is more complex than Craps so should be 99.60-99.70%. 0.1% from heaven. Alas, still great game, where you need to earn an almost equal chance against the casino.

Quote:MattUKBecause like every wacko I like doing things to completion, here's the 1-page best strategy ready to print. Two, actually. One most user-friendly, the other with EV. It's in open file format so easy to copy and change. Maybe it's worth releasing urbi et orbi.

https://drive.google.com/file/d/1tQuGA70dPMElB_UH3pAGr9fHayfYY59z/view?usp=sharing

Very nice work! May I use this on Wizard of Odds? I'm happy to give proper attribution, of course. Just let me know how you would like to be known.

I am about 80% done with a program to analyze the game and it is looking like it may run rather slowly if the player ever re-rolls four dice. Never a need to re-roll all five, obviously. It seems clear from your strategy that the player never rolls four, but just to be safe, can you confirm that?

I think it's rather clear that

(1) the only way to improve having Four of a kind is to go for Five of a kind,

(2) you must roll, or you'll waste the opportunity to improve from 2.500 to 4.000

(3) it's FAR easier to match the fifth die to the Four of a kind (match the tail to a dog) than four dice to the fifth (match the dog to a tail).

The usefulness of this reasoning is not exhaustive.

And I bet if the game gets any traction at all, they will add those progressives.

Quote:ThatDonGuyThe RTP is 99.5376324%

Not to say this is wrong, but the following table shows what I'm getting. My EV equates to an RTP of 96.7941%. Any thoughts about something I may not be seeing. I've gone over my code over and over, rewriting it, and I always get the same bottom line.

The only thing I can think of is my program never re-rolls 4 or all 5 dice. Your strategy says not to do that anyway. I'm stumped.

Event | Pays | Combinations | Probability | Return |
---|---|---|---|---|

Five of a kind on first roll | 9 | 362,797,056 | 0.000772 | 0.006944 |

Five of a kind | 3 | 18,570,954,240 | 0.039497 | 0.118491 |

Four of a kind | 1.5 | 86,360,256,000 | 0.183673 | 0.275509 |

Straight | 1 | 33,457,950,720 | 0.071159 | 0.071159 |

Full house | 0 | 94,383,221,760 | 0.200736 | 0.000000 |

Loser | -1 | 237,049,804,800 | 0.504163 | -0.504163 |

Total | 470,184,984,576 | 1.000000 | -0.032059 |

Quote:charliepatrickI agree with the paying figures except I get 39 907 676 160 straights, so it looks as if you've lost some after the 1st roll with hands like KKQJT (i.e. Pair and Easy Straight) 622080 and half the no pair straight draws (AKQJ9 etc) (414720/2).

Thanks Charlie. Stupid question -- the video shows the game being played with dice numbered 1 to 6. However, you refer to poker dice (9 to A). This would't make any difference would it? Aces high only, right?

My program just loops and makes the best of the 26 plays that re-roll 0 to 3 dice (never 4 or 5). I coded no strategy decisions in it.

What I originally used was a simple spreadsheet that just looked at EVs and percentages of various options. It looked at the chances of drawing different types of hands given your current set of dice. But it was easy to work out how many combinations this meant at various stages given the strategy. Luckily, unlike poker with cards, there's no memory of the discards since the dice are rerolled.

I would like to starting writing a page on the game and thinking about posting both of our figures for now.

***

RETURN TO PLAYER

Return To Player (RTP): 99.54%

The expected return is the amount we pay out to players relative to the amount of wagering on the game.

For example, if we take £100 of wagers we will, on average, pay out £99.54 of wins. This return is based on the player making the best choices.

Quote:WizardCharlie, not to doubt you, but has anyone else verified your number? I have seen MattUK quoting it, but Matt, can you confirm it?

I would like to starting writing a page on the game and thinking about posting both of our figures for now.

I get 99.5376324% also.

Quote:CrystalMathI get 99.5376324% also.

As did I.

It takes my program about two hours to cycle. Hope to have good news soon. My EV's on some sample hands agrees with those of MattUK.

Thanks again everybody for your help!

Quote:DJTeddyBearNice job, but I wish there was also a chart showing the possibility of each hand at the end of 3 rolls. Kind of important when figuring the house edge.

Here's the complete breakdown:

362,797,096 Nice Dice

18,570,954,200 Five of a Kind (except Nice)

86,360,256,000 Four of a Kind

94,383,221,760 Full House

39,907,676,160 Straight

86,556,211,200 Three of a Kind

70,073,579,520 Two Pair

63,892,592,640 One Pair

10,077,696,000 No Pair/Straight (12346, 12356, 12456, 13456)

Event | Pays | Combinations | Probability | Return |
---|---|---|---|---|

Five of a kind on first roll | 9 | 362,797,056 | 0.000772 | 0.006944 |

Five of a kind | 3 | 18,570,954,240 | 0.039497 | 0.118491 |

Four of a kind | 1.5 | 86,360,256,000 | 0.183673 | 0.275509 |

Straight | 1 | 39,907,676,160 | 0.084877 | 0.084877 |

Full house | 0 | 94,383,221,760 | 0.200736 | 0.000000 |

Loser | -1 | 230,600,079,360 | 0.490445 | -0.490445 |

Total | 470,184,984,576 | 1.000000 | -0.004624 |

They could have had a much more attractive pay table if a straight paid more than four of a kind. In the game of Yahtzee, a large straight pays 40, while a four of a kind pays only the "sum of the dice," which would average 17.5 assuming no strategy to go for the higher four-of-a-kinds.

Quote:MattUKPersonally, I like to think that the chance to get Nice Dice is 1/1296 and the return is 9/1296 = 1/144. On your table, it's extended by as much as 6^7. Still true, but devoid of much of elegance. For the same reason, the total number is 6^15 and all values can be divided by at least 6^7 to get more manageable figures.

The reason the total is 6^15, is that you have six dice that can be rolled three times. Much like in video poker many of my return tables total to 5*combin(52,5)*combin(47,5), although there is usually a common denominator to each line item.

After all the work I put into my program, I prefer to leave the raw results as they are.

Not sure who is wrong, but someone is!

Quote:rsactuaryYour answers differ than Don Guy's by 40 in the 5 of a kind (not nice) and loser categories.

Not sure who is wrong, but someone is!

470,184,984,576 / 1296 shows that Wizard is right.

Event | Combinations | Pays | Contribution |
---|---|---|---|

Nice Dice | 362 797 056 | 10 | 3 627 970 560 |

Fiver | 18 570 954 240 | 4 | 74 283 816 960 |

Quads | 86 360 256 000 | 3 | 215 900 640 000 |

FH | 94 383 221 760 | 1 | 94 383 221 760 |

Straight | 39 907 676 160 | 2 | 79 815 352 320 |

Trips | 86 556 211 200 | ||

Two Pair | 70 073 579 520 | ||

Pair/EasyStr | 14 511 882 240 | ||

Pair/InStr | 40 310 784 000 | ||

Pair/NoStr | 9 069 926 400 | ||

Inside Str | 10 077 696 000 | ||

TOTAL | 470 184 984 576 | 468 011 001 600 | |

Any Pair | 63 892 592 640 | RTP: | 99.537 632 411% |

Quote:rsactuaryYour answers differ than Don Guy's by 40 in the 5 of a kind (not nice) and loser categories.

Not sure who is wrong, but someone is!

My "loser numbers" add up to the Wizard's.

You're right about the "not nice 5 of a Kind" number; I calculated the total 5 of a Kind count and then subtracted the wrong number of Nice Dice rolls. Notice that my Nice Dice number is off by 40 as well.

Thanks in advance.