Stats question

[GreedySmurf]

Relating to a board game I play, I’m trying to calculate a fairly hefty conditional statistics exercise and my 30 something year old high school statistics classes aren’t cutting it. :o

I essentially have the choice of two different bonuses;
Bonus A - provides a +1 bonus on a contested roll at the start of the game, 2d6 (ie 2-12).
- If your opponent wins, there is no further bonus for you,
- If you win, then on ~three occasions (I’ll use 3 for the sake of the exercise, but the actual number is
variable), there will be a contested roll as follows:
* 7/11 - no impact
* 2/11 - a contested d6 roll with +1 to your roll
* 2/11 - a contested d3 roll with +1 to your roll.

Bonus B - provides a much more specific bonus. The initial contested roll is unaffected, your bonus applies regardless of the outcome of that contest.
- the same three contested rolls as above over the course of the game, only this time
* 9/11 - no impact
* 2/11 - contested d3 roll with +1 to your roll
Usually I can kludge up a matrix of sorts to brute force a simple stats answer, but the thing that is throwing of my attempts to solve this is the contested nature of the rolls. In that there is not just a single unmoving yes/no number for success. How do you work that out?

Any maths whizzes feel like at least pointing me in the right direction?

[Chronos]

We need more info. First of all, this initial contested roll: I take it that both players roll 2d6, and whoever rolls higher, gets some benefit? Is there some benefit other than the effect on the later contests (such as, whoever wins takes the first turn)? What happens if the contested 2d6 roll is tied? A re-roll? If it’s a re-roll, does the bonus apply on the re-roll also?

On the later rolls, is “no impact” what you want, or what you don’t want? That is to say, is it the equivalent of automatically succeeding on the subsequent contested roll, or the equivalent of automatically failing on it?

It might help to attach some names to these various situations where one would roll. The names won’t change the statistics, of course, but it’ll make it easier to talk about it.

[GreedySmurf]

@Chronos

We need more info. First of all, this initial contested roll: I take it that both players roll 2d6, and whoever rolls higher, gets some benefit? Is there some benefit other than the effect on the later contests (such as, whoever wins takes the first turn)? What happens if the contested 2d6 roll is tied? A re-roll? If it’s a re-roll, does the bonus apply on the re-roll also?

On the later rolls, is “no impact” what you want, or what you don’t want? That is to say, is it the equivalent of automatically succeeding on the subsequent contested roll, or the equivalent of automatically failing on it?

It might help to attach some names to these various situations where one would roll. The names won’t change the statistics, of course, but it’ll make it easier to talk about it.

Thanks Chronos.

I was trying not to get too bogged down but point taken.

Bonus A is Fan Factor
Bonus B is Cheerleaders

The game is a football simulation, and the initial 2d6 roll is indeed made by both players, and is a Fame roll, ie it tells you how many fans of your football team have come to watch. The higher result gets a +1 bonus to some of the later rolls (kick off). A tie means both players get the +1 bonus. The Bonus A is fan factor representing extra rusted on fans and gives you a +1 on the Fame roll.

The secondary set of events are kick-offs and introduce an extra random element to the game, where 2d6 are rolled and a table is consulted for the effect, things like a high kick, or particularly effective cheerleaders giving your team a boost.

On the kick off table, there are
A) Seven results which have various affects, but cannot be affected by the Fame, nor cheerleaders bonuses.
B) Two results which require a contested d6 roll, the winner gets a boost. The Fame +1 result is added to these rolls.
C) One result which requires a contested d3 roll, the winner gets a boost. The Fame +1 result is added to this rolls. and
D) One result requires a contested d3 roll, the winner gets a boost. Both the Fame +1 result and/or the Cheerleader +1 (Bonus B) is added to this roll. (I stuffed up my original description)

The desirable result is to win those four contested rolls if they come up. The other seven outcomes are set as to who benefits so which bonus I chose is immaterial.

So I guess I’m asking for help on deciding, statistically, is it better to take:
A) +1 Fan factor for a small influence on the Fame roll, (+1 on 2d6) for a chance at bonuses to win 4 of 11 options on the kick off table, or
B) +1 Cheerleaders for a guaranteed relatively large bonus (+1 on a d3) on only a single result of the 11 possible kick off table outcomes.
Given that the kick off table will on average be rolled on three times per game.

I hope that’s a bit easier to understand. :o

[Chronos]

One thing I’ll point out right away that the outcomes on the kickoff table aren’t 7/11, 2/11, 1/11, and 1/11. All of those should be something out of 36. An event that happens on a 7 on the kickoff table is a lot more likely (and thus, should carry more weight in our calculations) than one that happens on a 2 or 12 on the kickoff table.

[Buck_Godot]

@GreedySmurf

Thanks Chronos.

I was trying not to get too bogged down but point taken.

Bonus A is Fan Factor
Bonus B is Cheerleaders

The game is a football simulation, and the initial 2d6 roll is indeed made by both players, and is a Fame roll, ie it tells you how many fans of your football team have come to watch. The higher result gets a +1 bonus to some of the later rolls (kick off). A tie means both players get the +1 bonus. The Bonus A is fan factor representing extra rusted on fans and gives you a +1 on the Fame roll.

The secondary set of events are kick-offs and introduce an extra random element to the game, where 2d6 are rolled and a table is consulted for the effect, things like a high kick, or particularly effective cheerleaders giving your team a boost.

On the kick off table, there are
A) Seven results which have various affects, but cannot be affected by the Fame, nor cheerleaders bonuses.
B) Two results which require a contested d6 roll, the winner gets a boost. The Fame +1 result is added to these rolls.
C) One result which requires a contested d3 roll, the winner gets a boost. The Fame +1 result is added to this rolls. and
D) One result requires a contested d3 roll, the winner gets a boost. Both the Fame +1 result and/or the Cheerleader +1 (Bonus B) is added to this roll. (I stuffed up my original description)

The desirable result is to win those four contested rolls if they come up. The other seven outcomes are set as to who benefits so which bonus I chose is immaterial.

So I guess I’m asking for help on deciding, statistically, is it better to take:
A) +1 Fan factor for a small influence on the Fame roll, (+1 on 2d6) for a chance at bonuses to win 4 of 11 options on the kick off table, or
B) +1 Cheerleaders for a guaranteed relatively large bonus (+1 on a d3) on only a single result of the 11 possible kick off table outcomes.
Given that the kick off table will on average be rolled on three times per game.

I hope that’s a bit easier to understand. :o

You haven’t told us what happens if an opposed roll comes up a tie, but if we assume that you re-contest ties, then based on your corrected description I can tell you right now that option A is better even without calculation.

The initial contested roll will have a 50% chance of giving you a bonus (you and your opponent are symmetric)

If you get that bonus you get a 2/11 of having a +1 bonus on a d6 roll, and a 2/11 of having a +1 bonus on a d3 roll. So overall you have a 1/11 of getting a +1 bonus on a d6 roll and a 1/11 of getting a +1 bonus on a d3 roll.

If you choose option B all you get is a 1/11 of getting a +1 bonus on a d3 roll.

So Option A is better than option B by an additional 1/11 of getting a bonus on a 1d6 roll.

This is only for one kickoff, but in terms of the overall expected number of wins, expanding the number to 3 or 4 doesn’t really change anything.

If you lose the bonus on a tie, then it gets slightly worse for option A, but I think its still better.

Now that I answered your main question I can also tell you about opposed rolls.

In general for single die oppose rolls. If you have an N sided die, there are N^2 possible rolls. of these N-k will be different by k or by -k.

So in an opposed d3 roll,
3/9 will be different by 0,
2/9 will be different by +1,
2/9 will be different by -1,
1/9 will be different by -2,
1/9 will be different by +2.

[Buck_Godot]

@GreedySmurf

Thanks Chronos.

I was trying not to get too bogged down but point taken.

Bonus A is Fan Factor
Bonus B is Cheerleaders

The game is a football simulation, and the initial 2d6 roll is indeed made by both players, and is a Fame roll, ie it tells you how many fans of your football team have come to watch. The higher result gets a +1 bonus to some of the later rolls (kick off). A tie means both players get the +1 bonus. The Bonus A is fan factor representing extra rusted on fans and gives you a +1 on the Fame roll.

The secondary set of events are kick-offs and introduce an extra random element to the game, where 2d6 are rolled and a table is consulted for the effect, things like a high kick, or particularly effective cheerleaders giving your team a boost.

On the kick off table, there are
A) Seven results which have various affects, but cannot be affected by the Fame, nor cheerleaders bonuses.
B) Two results which require a contested d6 roll, the winner gets a boost. The Fame +1 result is added to these rolls.
C) One result which requires a contested d3 roll, the winner gets a boost. The Fame +1 result is added to this rolls. and
D) One result requires a contested d3 roll, the winner gets a boost. Both the Fame +1 result and/or the Cheerleader +1 (Bonus B) is added to this roll. (I stuffed up my original description)

The desirable result is to win those four contested rolls if they come up. The other seven outcomes are set as to who benefits so which bonus I chose is immaterial.

So I guess I’m asking for help on deciding, statistically, is it better to take:
A) +1 Fan factor for a small influence on the Fame roll, (+1 on 2d6) for a chance at bonuses to win 4 of 11 options on the kick off table, or
B) +1 Cheerleaders for a guaranteed relatively large bonus (+1 on a d3) on only a single result of the 11 possible kick off table outcomes.
Given that the kick off table will on average be rolled on three times per game.

I hope that’s a bit easier to understand. :o

You haven’t told us what happens if an opposed roll comes up a tie, but if we assume that you re-contest ties, then based on your corrected description I can tell you right now that option A is better even without calculation.

The initial contested roll will have a 50% chance of giving you a bonus (you and your opponent are symmetric)

If you get that bonus you get a 2/11 of having a +1 bonus on a d6 roll, and a 2/11 of having a +1 bonus on a d3 roll. So overall you have a 1/11 of getting a +1 bonus on a d6 roll and a 1/11 of getting a +1 bonus on a d3 roll.

If you choose option B all you get is a 1/11 of getting a +1 bonus on a d3 roll.

So Option A is better than option B by an additional 1/11 of getting a bonus on a 1d6 roll.

This is only for one kickoff, but in terms of the overall expected number of wins, expanding the number to 3 or 4 doesn’t really change anything.

If you lose the bonus on a tie, then it gets slightly worse for option A, but I think its still better.

Now that I answered your main question I can also tell you about opposed rolls.

In general for single die oppose rolls. If you have an N sided die, there are N^2 possible rolls. of these N-k will be different by k or by -k.

So in an opposed d3 roll,
3/9 will be different by 0,
2/9 will be different by +1,
2/9 will be different by -1,
1/9 will be different by -2,
1/9 will be different by +2.