A basic article on the use of probabilities in tactical decisions

March 29, 2005

Edited by John Olsen from a discussion on the SL list based on contributions from:

John Blazel, Paul Kuty, Joe Deuter, and Alan Yngve

Squads are in adjacent hexes, do you have them each fire separately at a target? Alternatively, do you have them combine their firepower into one attack? This question, recently posed by John Blazel (JB) to the SL list sparked a mathematical discussion. The purpose of this article is to not only recap and summarize the discussion but also to provide the reader with a mathematical foundation for future tactical decisions.

On February 22, 2005, JB asked:

"How would a person determine the chance of an event occurring at least once in two or more rolls? For example, if there is a line of 3 units it would be helpful to know whether to shoot them as one fire group or to shoot all 3 separately."

The mathematical answer to this question is pretty straightforward. Paul Kuty (PK) correctly pointed this out by stating:

"The chance of something happening at least once is 1 ? (the probability of it never happening)? which is back to the formula for probability of independent events?

2x4-6-7 shoot at a squad in a wooden building.

The chance to hit as one group (8IFT, +2 wood building) = 42%"

Before we continue, a discussion on calculating probabilities is in order. First, let?s clarify PK?s example. On the 8IFT, an 8 or less is needed to score a hit. With a +2 modifier, the required dice roll is reduced to a 6 or less. Next, we need to calculate the probability of rolling a 6 or lower with two dice. How is this accomplished?

Probability is the statistical likelihood of a certain event taking place. Probability is calculated by dividing the number of desired outcomes by the total possible outcomes. The probability of any number of events occurring is the sum of the probabilities of the individual events. The probability of two events occurring in sequence is the probability of the first event multiplied by the probability of the second event. In our example, we are rolling two six sided dice. The two dice have 36 different outcomes as outlined below.

As can be seen from the above chart, 15 of these outcomes will result in a 6 or lower. The probability is calculated as 15 desired outcomes/36 total possible outcomes, or 41.7%. The advancephase.com website contains this chart of dice roll probabilities. PK continues:

The chance of at least one is 1 ? (chance of all not occurring), or

the chance of both missing:

4IFT, +2 wood building = 83%

4IFT, +2 wood building = 83%

Probability (A) * Probability (B) = .83 *.83 =69% (rounded)

So, the chance of hitting the target is 31% (1 - .69).

Again, let?s make sure that this is clear. The easiest way to determine if an event occurs at least once is to determine the likelihood of the event never occurring and subtracting that number from one. Once we know the probability of both squads missing the target, it is then easy to calculate the probability of at least one squad scoring a hit. It is faster to determine the probability in this manner than trying to determine all of the possible probabilities of scoring a hit (i.e. squad A hits, squad B misses; squad A misses, squad B hits; both squad A & B hit).

Assuming the two squads are now going to attack separately, they would each attack on the 4IFT. This would require a roll of 6 or less. Since the target is still in a wooden building, the +2 modifier means that a 4 or less is needed. Of the 36 possible outcomes from rolling two dice, Table 1 illustrates that 6 outcomes would result in a roll of 4 or less. This translates into a probability of 6/36, or 16.7%. The probability of missing the target is 1 ? the chance of hitting, in this case 1 - .167 = .833, or 83%. The second squad has the same odds as the first which means the second squad also has an 83% chance of missing. Determining the probability of both squads missing is calculated by multiplying the probability of each individual event. PK demonstrated that this probability would be 69%. As also illustrated by PK, the chance of hitting the target at least once would then be 31%. In this example, the chance of hitting the target is better (42%) by combining the firepower of the two squads, rather than firing separately (39%). PK concludes his commentary:

"If it is a positive modifier to the roll it is best to use a large group? with a 0 or less modifier it is better to break it up into smaller attacks?"

Is this true? This is an axiom that has been widely adopted by the SL community. It does not take any mathematical prowess to understand and is straightforward and easy to remember. However, does the math bear this out? Keep reading to see what some other members of the SL community have to say on this subject. Joe Deuter (JD) also responds to JB?s question:

"8IFT, +0 attack upon a 7 morale level squad with no leader

NE Odds: 27.8%

MC Odds: 72.2%

KIA Odds: 8.3%

Squad KIA Odds: 8.3%

Squad Break Odds: 46.0%

But if you put in the moving in open modifier?

8IFT, -2 attack upon a 7 morale level squad with no leader

NE Odds: 8.3%

MCOdds: 91.7% (10 or less)

KIA Odds: 27.8% (5 or less)

Squad KIA Odds: 27.8%

Squad Break Odds: 67.9%

JD then responded to PK?s comments:

"This is not quite the whole picture because you can in fact score a hit on your attack roll but your opponent passes a subsequent MC roll for an overall result of NE.

Examples?

1). 8IFT, +2 wood building on a 4-4-7 squad

NE odds: 58.3%

MC odds: 41.7% (6 or less)

KIA odds: 0.0% (Impossible!)

Squad KIA: 0.0%

Squad breaks: 23.1%

And?

2). 4IFT, +2 wood building on a 4-4-7 squad

NE odds: 83.3%

MC odds: 16.7% (4 or less)

KIA odds: 0.0% (Impossible!)

Squad KIA: 0.0%

Squad breaks: 8.3%

Therefore?

1). (1-.231) = 76.9% Overall chance of NE

2). (1-.083)^2 = 84.1% Overall chance of NE

In his posts, JD is pointing out the importance of actually affecting the enemy unit (either breaking or eliminating) rather than just scoring a hit on the IFT. Every player should be more concerned with the likelihood (aka the probability) of breaking/destroying the target, rather than only hitting the target.

PK responds:

"Since you are trying to determine the odds to BREAK a unit? not just for something negative to happen?In order to break a 7 morale level squad (@8IFT, +0) you must roll a 4,5,6,7,8 in order to make a MC. A 2,3 are a KIA and thus not in the result set.

Therefore your comment

8IFT, +0 attack upon a 7 morale level squad with no leader

NE Odds: 27.8%

MC Odds: 72.2%

KIA Odds: 8.3%

Should read

NE Odds: 27.8%

MC Odds: 63.9% (72.2% to hit ? 8.3% KIA)

KIA Odds: 8.3%

Which leads the premise if you want to BREAK a squad you must roll a 4,5,6,7,8 and then a number based on that result. So?

Case B 8IFT, +0

drdr%break %breaking22.80%0.0%0.0%KIA35.60%0.0%0.0%KIA48.30%72.2%5.99%2MC511.10%72.2%8.01%2MC613.90%58.3%8.1%1MC716.7%58.3%9.74%1MC813.90%41.7%5.80%MC72.3%37.64%So the odds of a 7 morale level squad BREAKING on 8IFT +0 is 37.64%

The odds of KIA are 8.3%

The odds of something bad happening are 45.94% (37.64+8.3)

Anyway that?s my $.02"

JD confirms PK?s conclusions:

"Yes, I include the KIA result as a broke as well. Technically the broke result is ?Broke or worse.?

That was a lot to swallow. Let?s try to make this a little more digestible. Remembering that our goal is to affect the enemy unit (i.e. break or KIA), we need to determine the probability of achieving that outcome. Determining the probability of a KIA is relatively straight forward. In the above example, where we are utilizing 8IFT with no modifiers, a dice roll of either 2 or 3 will result in a KIA. As detailed in Table 1 above, there are 3 ways to roll a 2 or 3. This translates into a probability of 3/36 = 8.3%. The tougher issue is determining the likelihood of the target breaking due to a morale check. How do we know whether the target will break or not? The same probability analysis in determining whether the target is hit can also be used to determine whether the target will break. PK, in his response to JD, provided a chart illustrating the probability of the target breaking. Let?s examine this chart in more detail:

Case B 8IFT, +0

drdr%break %breaking22.80%0.0%0.0%KIA35.60%0.0%0.0%KIA48.30%72.2%5.99%2MC511.10%72.2%8.01%2MC613.90%58.3%8.1%1MC716.7%58.3%9.74%1MC813.90%41.7%5.80%MC72.3%37.64%So the odds of a 7 morale level squad BREAKING on 8IFT +0 is 37.64%

The odds of KIA are 8.3%

The odds of something bad happening are 45.94% (37.64+8.3)

The first column (dr) is the IFT dice roll. The second column (dr%) is the probability of that particular number being rolled. The third column is the most telling. For that particular roll, the third column (break %) gives the probability of the unit breaking as a result of that roll. An example will most clearly illustrate how this probability is calculated. Assume that a 4 is rolled. A 4 will result in a 2MC. This means that the target (with a 7 morale level) will have to roll a 5 or lower in order to pass the morale check. A roll of 6 or higher will result in the target unit breaking. PK indicates that the target will break with a probability of 72.2%. There are 26 ways of rolling a 6 or greater. This translates into a probability of 26/36 = 72.2%. So, now we have two events that we want to know the likelihood of occurring: rolling a 4 (8.3%) followed by a roll of 6 or greater (72.2%). The product of these two individual probabilities will provide the probability of both events occurring. This is detailed in PK?s fourth column (breaking). The same calculation is done for the other possible to hit dice rolls and then adding the individual probabilities together to determine the likelihood of any one of those events taking place.

The bottom line is that in this example the target will break/KIA with a probability of 45.94%

Alan Yngve (AY) added this:

"It is possible (and some do this) to have a probability chart beside them when they play. But especially in FTF games, having a simpler "rule of thumb" is incredibly helpful. I suspect that the guideline listed above is one of the best. Use it when you are trying to decide whether to create a multi-hex firegroup, or not.

For those interested in a published article on this topic, try to find a copy of "Basic Arms and the Man" by Mark Swanson in issue 18-5 of The (AH) General.

The other thing to consider is what are you trying to measure?

If one side has a three-squad stack that he wants to move against opposition. What is important?

--Needs to get all three squads through safely?

--Needs to get at least one squad through safely?

--Needs to avoid a KIA?

All of these situations will generate different probabilities and (perhaps) different tactics!

So, enjoy Squad Leader, there?s a lot there to think about!"

I would also add to the reading list issue 14-5 of the General. Bob Medrow does an excellent job of discussing these aspects of Squad Leader. As Alan points out, due to the nature of the IFT, probabilities play an important factor in many different tactical decisions that a player must make during the course of a game. And, at least to me, these decisions are not dissimilar to the decisions that battalion commanders needed to make during actual combat operations that Squad Leader simulates.

Now that we know how the math works, let?s examine JB?s question a little further by doing some examples to see if the axiom stated by PK and endorsed by AY actually is true.

Example 1:

Three 4-6-7 squads in three adjacent hexes are going to fire on the same target hex, a 4-4-7 unit in a wooden building. Do they fire together, or separately?

a). separate attack. Each squad would have the following probabilities:

4IFT, +2

To hit: 16.7%

To kill: 0.0%

To break: 8.4%

Target unaffected: 91.6% (1-8.4%)

The To break probability was calculated in the same manner that PK demonstrated above:

dice roll | result | chance of rolling | chance of breaking | break % |

2 | 1MC | 2.80% | 58.3% | 1.6% |

3 | 1MC | 5.60% | 58.3% | 3.3% |

4 | MC | 8.30% | 41.7% | 3.5% |

Total break probability | 8.4% |

The two other units would have the same probability of breaking the enemy unit. In order determine the likelihood of at least one of the attacking units breaking the target would be 1 ? the likelihood of all three leaving the target unaffected:

1- (.916 * .916 *.916) = **23.1%**.

b). Combined attack. The firegroup would have the following probabilities:

12IFT, +2

To hit: 58.3%

To kill: 0.0%

To break: 33.9%

Target unaffected: 66.1%

To break:

dice roll | result | chance of rolling | chance of breaking | break % |

2 | 3MC` | 2.80% | 83.3% | 2.3% |

3 | 2MC | 5.60% | 72.2% | 4.0% |

4 | 2MC | 8.30% | 72.2% | 6.0% |

5 | 1MC | 11.10% | 58.3% | 6.5% |

6 | 1MC | 13.90% | 58.3% | 8.1% |

7 | MC | 16.7% | 41.7% | 7.0% |

Total break probability | 33.9% |

In this example, a). would have a 23.1% probability of breaking the enemy, while b). has a 33.9% probability. It is obvious a combined attack would have a better chance of success. The following examples illustrate the same situation with different terrain effects modifiers. The math has been omitted to keep this article from getting any longer.

Example 2:

Three 4-6-7 squads in three adjacent hexes are going to fire on the same target hex, a 4-4-7 unit in an open ground hex. Do they fire together, or separately?

a). separate attack. Each squad would have the following probabilities:

4IFT, +0

To hit: 41.7%

To kill: 2.8%

To break: 21.1%

Target unaffected: 76.1% (1 - 21.1% - 2.8%)

Chance of at least one attack resulting in a break (or better):

1 ? (.761 * .761 * .761) = **55.9%**

b). Combined attack. The firegroup would have the following probabilities:

12IFT, +0

To hit: 83.3%

To kill: 8.3%

To break: 47.3%

Target unaffected: 44.4%

Break or KIA of the target: **55.6%** (8.3% + 47.3%)_

It is interesting to note that with a 12IFT, the chance of hitting the target is significantly better than any individual attack at 4IFT. However, the chance of affecting the enemy is slightly better with three separate attacks (55.9%) versus one attack (55.6%).

Example 3:

Three 4-6-7 squads in three adjacent hexes are going to fire on the same target hex, a 4-4-7 unit moving in an open ground hex. Do they fire together, or separately?

a). separate attack. Each squad would have the following probabilities:

4IFT, -2

To hit: 72.2%

To kill: 16.7%

To break: 31.6%

Target unaffected: 51.7% (1 ? 31.6% - 16.7%)

Chance of at least one attack resulting in a break (or better):

1 ? (.517 * .517 * .517) = **86.2%**

b). Combined attack. The firegroup would have the following probabilities:

12IFT, -2

To hit: 97.2%

To kill: 27.8%

To break: 47.3%

Target unaffected: 24.9%

Break or KIA of the target: **75.1%** (27.8% + 47.3%)

The chance of affecting the enemy unit using separate attacks is better (86.2%) than with one combined attack (75.1%). It is important to note a couple of items that this analysis does not consider. First, it does not differentiate between KIA and breaking a target unit. Depending upon the circumstances and victory conditions of any scenario, this may influence your attack decision. Second, while there is a greater chance of achieving a KIA with a larger firegroup, the same result can be achieved if multiple small attacks result in a double break. Third, the analysis does not consider the impact of utilizing smaller attack groups to provide greater flexibility in overall tactical decisions. If you decide to separate the attack and score a break on the first combat resolution, the other units can now fire at another target or are free to move.

To close, PK?s comment on combining fire with positive modifiers and separating fire with negative modifiers is certainly borne out by the mathematical analysis. But as AY pointed out, the game of Squad Leader requires many other tactical decisions. Every decision will have a mathematical affect on the outcome of the game. Good luck and good hunting.