Dice odds and re-rolls

[J_Bone]

Hi guys... This is one for the more maths oriented among the forum members. I was wondering how to figure out the dice odds when it comes to thinking about one roll after the other and how re-rolls affect this. I am pretty good with normal D6 odds (such as the odds of rolling a 12 with two dice or whatever) but I have not been able to find ways of calculating how re-rolls and multiple rolls work in order to affect the odds of the outcome of multiple actions.

For example, I require a 5+ roll to make a pass but if I make a 2+ roll for a GFI I can make it a 4+ roll. Is a 2+ and then a 4+ better odds than a 5+? I would assume it is but how is this calculated?

Also, how do re-rolls affect odds? I have seen people mentioned a 50% chance of success (4+ on a D6 for instance) with a re-roll becoming a 75% chance of success but I was curious as to the maths behind this.

[J_Bone]

The assumption I have been working on for the independent rolls is that it would work in much the same way as calculating the odds for a double 6 on 2D6 in that you just times the odds of each event by the other so 1/6 x 1/6 becomes 1/36. So the 2+ then a 4+ becomes 5/6 x 3/6 = 15/36 = 2.4/6 so this action is 0.4/6 more reliable than a 5+ roll.

[Jimmy Fantastic]

This is great - http://www.elyoukey.com/sac/

[lunchmoney]

To take your example the 5+ roll has a 2/3 chance. To get the percentage odds simply do that exact equation: 2 divided by 3. This will give you a 0.xx number. The xx is the percentage, in this case 66.6%.
The 2+ roll is 5/6, that gives 83.3%, the subsequent 4+ roll is ½ or 50%. To combine them to the total odds, (5/6) x (1/2) = 0.416. Or 41.6%
So we can see the odds of doing a 5+ pass is better than a 2+GFI then 4+ pass.


As for your second example/question:
To do this one with maths in the same way as the first example is a little different as you do not need both rolls to succeed. It’s either/or. So, (1/2) x (1/2) actually gives the chance of failure = 25%. Which is the 75% chance of success


Does that make sense?

(I think that’s right, and I’m sure Dode (or someone) will be along to correct it.)

[J_Bone]

Hi Lunchmoney... I think you might be saying the same as I did in my second post in a different way. You appear to have made a slight error but worked in the same way... Because a 5+ definitely isn't a 2/3 chance of success. It's a 1/3 because it's 2/6. A 2/6 chance is 33.33% so a 2+ and then a 4+ at 41.6% is better. Is that right? Or have I lost my mind? Haha!

The re-roll calc makes sense though... I just need to digest it a little.

[J_Bone]

And thanks, Jimmy... That tool looks very useful!

[lunchmoney]

Hi Lunchmoney... I think you might be saying the same as I did in my second post in a different way. You appear to have made a slight error but worked in the same way... Because a 5+ definitely isn't a 2/3 chance of success. It's a 1/3 because it's 2/6. A 2/6 chance is 33.33% so a 2+ and then a 4+ at 41.6% is better. Is that right? Or have I lost my mind? Haha!
.

Yeah my bad. Mistyped it then didnt check when I pulled a calculator out. But the idea is sound

To take your example the 5+ roll has a 2/3 chance. To get the percentage odds simply do that exact equation: 2 divided by 3. This will give you a 0.xx number. The xx is the percentage, in this case 66.6%.
The 2+ roll is 5/6, that gives 83.3%, the subsequent 4+ roll is ½ or 50%. To combine them to the total odds, (5/6) x (1/2) = 0.416. Or 41.6%
So we can see the odds of doing a 5+ pass is better than a 2+GFI then 4+ pass.

Should read:
Pass roll 5+ = 1/3 = 33.3%, so the GFI and 4+ pass is better

[J_Bone]

Cool... I just need to wrap my head around the re-rolls now.

I am curious as to how only having one re-roll across multiple rolls works as well... Seems like that might be more complex than it would first appear.

[outcast]

There are people here who will be able to give you a much more accurate explanation of re-rolls but I find it easiest to calculate the odds of it failing to give me the odds of success.

For example, a 3+ dodge with a re-roll is basically a 4/6 on the first roll OR a 4/6 on the second.

I find it easier to work out an AND with fractions (multiplying them) than an OR so I calculate the odds of failing. So, to fail my 3+ dodge with a re-roll I would need a 2/6 followed by another 2/6 = 4/36 = 1/9. So to get my chances of success I just need to invert that fraction = 8/9 = 88.88%

I don't know if that makes ANY sense whatsoever but it does in my head!

[dode74]

You basically have to build a probability tree, the complexity of which will depend on the length of the sequence and any other skill rerolls in there. You then add the probabilities of success (or subtract the probabilities of failure from 1, which can also act as a checksum) to get your odds.

For example, with the GFI/4+ pass option vs the 5+ pass:

5+ Pass
Roll 1

• Pass roll: 1/3
  Fail roll: 2/3 -> go to roll 2

Roll 2

• Pass roll: 1/3
  Fail roll: 2/3

So, the odds of a pass are 1/3 for a straight success (leaving you with a reroll), with 2/3 * 1/3 = 2/9 for the reroll working, for a total success odds of 5/9.
The odds of a fail are 2/3 for the reroll to be used, followed by a 2/3 for the reroll to fail, for a total fail odds of 4/9.

GFI/4+
The GFI (2+)
Roll 1

• Pass: 5/6 (RR still available)
  Fail: 1/6 -> go to roll 2

Roll 2

• Pass: 5/6 (RR not available)
  Fail: 1/6

So, you could fail just on the GFI for 1/6 *1/6 = 1/36
You could pass but use the RR for 1/6 * 5/6 = 5/36
You could pass and still have the RR for 5/6 = 30/36
Checksum: 1/36 + 5/36 + 30/36 = 36/36, which works.

The Pass (4+)
Roll 1 - There is a 35/36 chance of you being here in the first place, with 5/36 chance of having no RR and 30/36 chance of having a RR.

• Pass: 3/6
  Fail: 3/6

So, the odds so far:
Failed at the GFI: 1/36 = 6/216
Pass and still have a reroll: 3/6 * 30/36 = 90/216
Pass and have no reroll: 3/6 * 5/36 = 15/216
Fail and still have a reroll: 3/6 * 30/36 = 90/216 -> go to roll 2
Fail and have no reroll: 3/6 * 5/36 = 15/216

Roll 2 - There is a 90/216 chance of being here in the first place (see above)

• Pass: 3/6
  Fail: 3/6

So, the overall odds:
Fail the GFI (1/6 * 1/6): 6/216
Pass the GFI using a RR (1/6 * 5/6), and succeed on the pass (3/6): 15/216
Pass the GFI using a RR (1/6 * 5/6), fail the pass (3/6): 15/216
Pass the GFI without a RR (5/6), fail the pass (3/6), RR and fail again (3/6): 45/216
Pass the GFI without a RR (5/6), fail the pass (3/6), RR and pass it (3/6): 45/216
Pass the GFI without a RR (5/6), pass the pass (3/6) = 90/216
Making the overall odds:
Fail: 66/216
Pass, but use a RR: 60/216
Pass and not use a RR: 90/216
Checksum: 66/216 + 60/216 + 90/216 = 216/216, which works.

Look complex? We've not even looked at the catch yet, or any skills such as pass or pro(!) ! This is why the Samba Action Calculator which Jimmy linked is so useful.

[lunchmoney]

See? I said Dode will be along and with a colour display as well

[Magictobe]

You should take in consideration that there is more then the odds itself.

With a failed GFI your players is down (with armour 7 teams, this can be a problem), with a failed pass your player is still on its feet.

[Heff]

And in any event the chance of failure is inverse to the importance of the roll.
A gfi to score fails 1/2 the time.
A GFI to equalise fails 5/6 of the time
and a last move death or glory overtime GFI to win the league will not only fail but kill your player, and the ball will bounce out and be thrown in at the feet of the opposing teams only unmarked AG5 thrower, who will the long bomb it to a player surrounded by four of yours, who will dodge away from the diving tackle tentacles prehensile tailed str 5 monster you have, hand off to another player who will catch it in three zones, and then dodge out and make two gfis to defeat you!

Cos that's a one in a million (Dode will now do the maths) and so will always work!

[daloonieshaman]

Heff
EXTREMELY UNPROBABLE
but yet....
it ALWAYS seems to work out that way

[dode74]

Cos that's a one in a million (Dode will now do the maths) and so will always work!

No need: Terry Pratchett did it already