### Risk strategy

Risk is game of strategy, without a good strategy you are bound to lose the game, however well you perform on individual turns. But when it comes down to it Risk is also a game of numbers, a game of statistics. When the player is looking at the board, deciding his next move, what he is trying to work out is whether or not he will be able to win each battle and, even if he does win, if he will be left weak and vulnerable afterwards.

Thus I present to you

**The Statistics of Risk and its Implications on Strategy**

Before we start it is important for me to briefly clarify a few definitions that I will use. An **attack** is one roll of the dice. A **battle** is a series of attacks that continue until the defending army is completely destroyed or the attacker decides to stop, for example if he were to be worn down to only having one army left. A **win**, when applied to a battle, is when the defender loses all of his pieces, when applied to an attack it is merely when the defender loses pieces and the attacker does not. A **loss**, when applied to a battle, is when the attacker stops attacking (if, for example, he were to only have one army left and thus can not attack) yet the defender still has at least one army in his territory. When applied to an attack it is when the attacker loses pieces and the defender does not. There is also the special case of a **draw**, this can only apply in attacks where the attacker is using 2 or 3 dice and the defender is using 2 dice, if both the attacker and the defender lose one army each then they have drawn.

I will assume that you know how attacks in Risk are conducted, but for those of you who have not played, or who are a bit fuzzy about it then here you go. The attacker can only attack from a territory which has more than one army in it. The attacker can roll up to three dice to attack, but the number of dice rolled must be less than the number of armies in the attacking territory. So if the attacker only has two armies in that territory then he can only roll one dice (if he has three armies then two dice, if four or more then all three die may be rolled). The defender may roll one or two dice. If the defender has one or two armies then only one die may be rolled, if he has three or more armies then he may defend with two dice.

Once the number of dice that may be rolled by both the attacker and the defender has been decided then both sides roll their dice. Once the dice have been rolled then the attackers highest rolled dice is matched with the defenders highest rolled dice, then the second highest with the second highest (assuming that either side attacked using more than one die). The values are then compared. If the attacker's dice has a higher value than the defender's then the defender loses an army. If the values are equal, or if the attacker's dice has a lesser value than the defender's then the attacker loses an army. This is done for each pair of dice. So it can be seen that, providing both attacker and defender roll at least two dice each, there are three possible outcomes: the defender loses two armies, the attacker loses two armies, or both attacker and defender lose an army. The last case happens when the attacker loses one of the pairs and wins the other pair, this case is a draw.

The first thing we must consider as we embark on this enterprise are the dice. The game is made or broke on the rolls of these simple cubes. There are three possible outcomes of any roll of the dice, the roll can be Won, Drawn or Lost. There are also six possible combinations of die rolls. So we should start of by looking at the probabilities of winning, losing or drawing any dice roll.

DEFENCE
WIN DRAW LOSE
| 1 | 2 | | 1 | 2 | | 1 | 2 |
A --+-------+-------+ --+-------+-------+ --+-------+-------+
T 1 | 0.417 | 0.174 | 1 | 0.000 | 0.000 | 1 | 0.583 | 0.826 |
T --+-------+-------+ --+-------+-------+ --+-------+-------+
A 2 | 0.660 | 0.228 | 2 | 0.000 | 0.324 | 2 | 0.340 | 0.448 |
C --+-------+-------+ --+-------+-------+ --+-------+-------+
K 3 | 0.802 | 0.372 | 3 | 0.000 | 0.336 | 3 | 0.198 | 0.293 |
--+-------+-------+ --+-------+-------+ --+-------+-------+

In these tables the rows are the number of dice that the attacker is rolling (from one to three) and the columns are the number of dice that the defender is rolling (one or two). So, for example, we can see that if the attacker is rolling one dice and the defender is rolling one dice then the probability of the attacker winning is 0.417 and the probability of the attacker losing is 0.583. Remember of course that the defender has a slight advantage since the defender wins the roll if the numbers are equal. Obviously in any case where either side is rolling less than two dice then a draw can not occur, thus having a probability of zero.

From these tables we should note several things. Firstly we should observe that the defenders ability to roll more than one die has a major effect on the attacker's chance of winning, in all cases more than halving the attacker's chance of winning. It should also be noted that, in all cases where the defender rolls two dice, the odds are not stacked in particular favor of the attacker. In all cases where the defender rolls two dice there is an at least 0.629 chance of the attacker losing at least one army, rising to 0.826 if the attacker was to attack with one army against the two.

These figures also demonstrate that a common strategy does not work. Many people, while defending their territory, will choose to defend with just one die (against three) under the preconception that it will reduce their loses. If we compare the probable outcomes of one attack, where the attack uses three dice and the defender uses two, to a series of two attacks in which the attacker uses three dice and the defender uses only one, we can see the following. In the first case of three against two the chances of the defender losing two armies is 0.372 and the chance of losing just one army is 0.336. In the second case of two attacks of three armies against just one army we can work out that the chances of the defender losing two armies (one in each battle) actually increases to 0.643. The chance of losing just one army does however drop to 0.318. If we combine these numbers then we can see that the chances of the defender losing *at least* one army is just 0.629 in the first case compared to an enormous 0.961 in the second case. This clearly shows that, in all cases, the defence should maximize their ability to defend by using both dice whenever they can, irregardless of the myth of "slowing down your loses".

These numbers clearly show us that in any attack losses are to be expected. Later we will return to this and look at the most likely number of losses that can be expected when fighting a battle. We will also look at the likely number of losses that a defender can be expected to sustain should the attacker lose the battle. But before we move on to that we shall first look at the probabilities of a battle being won or lost.

To generate the following figures I did the following. I started off by using the probabilities from the tables above. I made the assumption that, in all cases, both the attacker and defender will use the maximum number of dice that they can roll.

In any attack (where both attacker and defender are using at least two dice each) there are three possible outcomes: the attacker can win, lose or draw. Thus we need to look at the chances of any one of these three outcomes occurring. So we can work out the chance of the attacker winning a battle with *x* armies against the *y* armies of the defender as follows:

To start we must first define three functions, `p_ow(`*x*, *y*), `p_od(`*x*, *y*), `p_ol(`*x*, *y*). These give the probabilities of wining, drawing or losing an attack. Basicly they are just the contents of the dice tables given above. For the sake of completeness I will repeat them here.

p_ow(2, {1|2}) = 0.417 p_od(2, {1|2}) = 0.000 p_ol(2, {1|2}) = 0.583
p_ow(2, >2) = 0.174 p_od(2, >2) = 0.000 p_ol(2, >2) = 0.826
p_ow(3, {1|2}) = 0.660 p_od(3, {1|2}) = 0.000 p_ol(3, {1|2}) = 0.340
p_ow(3, >2) = 0.228 p_od(3, >2) = 0.324 p_ol(3, >2) = 0.448
p_ow(>3, {1|2}) = 0.802 p_od(>3, {1|2}) = 0.000 p_ol(>3, {1|2}) = 0.198
p_ow(>3, >2) = 0.372 p_od(>3, >2) = 0.336 p_ol(>3, >2) = 0.293

To make life simpler and easier to understand: `p_ow` means "probability of outcome win", `p_od` and `p_ol` are "outcome draw" and "outcome lose" respectively.

We can now define our function `p_win(`*x*, *y*). For the sake of ease of reading, `p_win` stands for probability win.

p_win(*x* > 0, 0) = 1
p_win(1, *y* > 0) = 0
while({x,y}>2)
p_win(*x*, *y*) = p_ow(*x, y*)p_win(*x, y* - 2) + p_od(*x, y*)p_win(*x* - 1, *y* - 1) + p_ol(*x, y*)p_win(*x* - 2, *y*)
else
p_win(*x*, *y*) = p_ow(*x, y*)p_win(*x, y* - 1) + p_ol(*x, y*)p_win(*x* - 1, *y*)

As you can see, this creates a nice recursive function which, conveniently enough, is nice and easy to code. Basiclly all the algorithm does is to create a probability tree and then sum the probabilities of each branch of the tree where the attacker wins. By plugging these algorithms into a nice C program I generated the following table of the probability of any attack winning with up to twenty armies against up to twenty armies. The hardcore risk players might want to print this out to refer to mid-game...

D E F E N C E
| 01 | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
01 |0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
02 |0.417|0.174|0.030|0.005|0.001|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
03 |0.802|0.588|0.239|0.144|0.056|0.033|0.013|0.008|0.003|0.002|0.001|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|0.000|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
04 |0.961|0.887|0.564|0.412|0.258|0.172|0.107|0.068|0.042|0.026|0.016|0.010|0.006|0.004|0.002|0.001|0.001|0.001|0.000|0.000|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
05 |1.000|0.971|0.737|0.593|0.429|0.317|0.221|0.156|0.106|0.073|0.049|0.033|0.021|0.014|0.009|0.006|0.004|0.003|0.002|0.001|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
06 |1.000|1.000|0.863|0.738|0.596|0.469|0.360|0.269|0.199|0.143|0.103|0.073|0.051|0.035|0.025|0.017|0.011|0.008|0.005|0.003|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
A 07 |1.000|1.000|0.922|0.835|0.716|0.604|0.489|0.391|0.303|0.234|0.175|0.131|0.096|0.070|0.050|0.036|0.025|0.018|0.013|0.009|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
T 08 |1.000|1.000|0.960|0.898|0.812|0.712|0.611|0.508|0.417|0.333|0.264|0.204|0.157|0.118|0.089|0.066|0.049|0.035|0.026|0.018|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
T 09 |1.000|1.000|0.978|0.939|0.875|0.799|0.708|0.617|0.523|0.438|0.358|0.290|0.230|0.181|0.140|0.108|0.082|0.062|0.046|0.034|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
A 10 |1.000|1.000|0.989|0.964|0.922|0.861|0.790|0.707|0.623|0.536|0.456|0.379|0.313|0.253|0.204|0.160|0.126|0.097|0.075|0.057|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
C 11 |1.000|1.000|1.000|0.980|0.950|0.908|0.850|0.784|0.707|0.630|0.548|0.473|0.399|0.334|0.274|0.224|0.180|0.144|0.113|0.089|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
K 12 |1.000|1.000|1.000|0.988|0.970|0.939|0.898|0.842|0.780|0.708|0.636|0.559|0.487|0.416|0.353|0.294|0.244|0.198|0.161|0.128|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
13 |1.000|1.000|1.000|1.000|0.982|0.962|0.930|0.889|0.836|0.777|0.710|0.641|0.568|0.500|0.432|0.370|0.312|0.262|0.215|0.177|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
14 |1.000|1.000|1.000|1.000|1.000|0.976|0.955|0.923|0.883|0.832|0.776|0.711|0.646|0.577|0.512|0.445|0.387|0.329|0.278|0.231|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
15 |1.000|1.000|1.000|1.000|1.000|0.987|0.971|0.948|0.916|0.877|0.828|0.775|0.713|0.652|0.586|0.522|0.457|0.398|0.339|0.286|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
16 |1.000|1.000|1.000|1.000|1.000|1.000|0.983|0.966|0.943|0.911|0.873|0.825|0.774|0.716|0.656|0.589|0.523|0.461|0.394|0.340|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
17 |1.000|1.000|1.000|1.000|1.000|1.000|1.000|0.979|0.961|0.938|0.906|0.868|0.823|0.774|0.714|0.645|0.586|0.516|0.445|0.387|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
18 |1.000|1.000|1.000|1.000|1.000|1.000|1.000|0.987|0.975|0.957|0.933|0.900|0.860|0.817|0.760|0.695|0.632|0.564|0.495|0.446|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
19 |1.000|1.000|1.000|1.000|1.000|1.000|1.000|1.000|0.985|0.971|0.952|0.924|0.891|0.848|0.801|0.737|0.664|0.606|0.549|0.459|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
20 |1.000|1.000|1.000|1.000|1.000|1.000|1.000|1.000|1.000|0.982|0.967|0.945|0.913|0.872|0.832|0.769|0.698|0.615|0.552|0.466|
----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+

So what can we learn from this?

Firstly we must note that this table merely gives the probability of winning, it tells us nothing about how many armies we are likely to have left afterwards, although we will be looking into this later. We should also note that it is very simple to work out the probability of winning if you also wish to have a certain number of armies left at the end, all you have to do is take as your attacking force the number of armies you have minus the number of armies you wish to have left (so if I have 20 armies and want to have 5 left at the end, then I take the probability of attacking with 15 armies). Lastly, the top row (attacking with one army) is supposed to be all zeros - you can't attack if you only have one army!.

One of the most obvious questions to ask of this data is "How many armies do I need to have good chance of winning?". To answer this using the data provided we must first define what is meant by a "good" chance of winning. For the sake of the reality of the game I am going to define a "good" chance as being two thirds. In my opinion any attack that has a two thirds chance of winning is "good". So let us look at the numbers provided above. The first thing that we can see (with the aid of a nice graphing program...) is that the ratio of attackers to defenders to ensure a chance of winning that is at least 0.66 follows an inverse exponential law. The ratio goes down from needing 3 attackers to defeat 1 defender, a ratio of 3, to a ratio of just 1.125 with 18 attackers against 16 defenders. We can also see a slightly surprising fact here. In order to have at least a 0.66 (two thirds) chance of beating a defender, the attacker needs to have merely two armies more than the defender has. Entirely counter-intuitive, but this result is entirely borne out by the numbers.

To take this further we must now look at a table of the most likely number of armies remaining after an attack. To generate these next two tables I used the same algorithm as previously but, instead of looking at the probabilities, I looked at the number of armies left at the end of each possible attack outcome. I then just worked out what the most likely number of armies was from this range.

The first table is the most likely number of armies the attacker will have left assuming that the attacking wins the battle. The second table is most likely number of armies that the defender will have remaining assuming that the attack fails.

D E F E N C E
| 01 | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
01 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 | 00 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 | 02 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
04 | 04 | 04 | 04 | 04 | 03 | 04 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
05 | 05 | 05 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 03 | 04 | 03 | 04 | 03 | 04 | 03 | 04 | 03 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
06 | 06 | 06 | 05 | 05 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
A 07 | 07 | 07 | 06 | 06 | 05 | 05 | 05 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
T 08 | 08 | 08 | 07 | 06 | 06 | 06 | 05 | 05 | 05 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
T 09 | 09 | 09 | 08 | 07 | 07 | 06 | 06 | 05 | 05 | 05 | 05 | 05 | 04 | 04 | 04 | 04 | 04 | 04 | 04 | 04 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
A 10 | 10 | 10 | 09 | 08 | 07 | 07 | 06 | 06 | 06 | 05 | 05 | 05 | 05 | 05 | 05 | 04 | 04 | 04 | 04 | 04 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
C 11 | 11 | 11 | 10 | 09 | 08 | 08 | 07 | 07 | 06 | 06 | 06 | 05 | 05 | 05 | 05 | 05 | 05 | 05 | 04 | 04 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
K 12 | 12 | 12 | 11 | 10 | 09 | 09 | 08 | 07 | 07 | 06 | 06 | 06 | 06 | 05 | 05 | 05 | 05 | 05 | 05 | 04 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
13 | 13 | 13 | 12 | 11 | 10 | 10 | 09 | 08 | 08 | 07 | 07 | 06 | 06 | 06 | 06 | 05 | 05 | 05 | 05 | 05 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
14 | 14 | 14 | 13 | 12 | 11 | 11 | 10 | 09 | 09 | 08 | 07 | 07 | 07 | 06 | 06 | 06 | 05 | 05 | 05 | 05 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
15 | 15 | 15 | 14 | 13 | 12 | 11 | 11 | 10 | 09 | 09 | 08 | 08 | 07 | 07 | 06 | 06 | 06 | 06 | 06 | 05 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
16 | 16 | 16 | 15 | 14 | 13 | 12 | 12 | 11 | 10 | 10 | 09 | 08 | 08 | 07 | 07 | 07 | 07 | 06 | 06 | 06 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
17 | 17 | 17 | 16 | 15 | 14 | 13 | 13 | 12 | 11 | 10 | 10 | 09 | 09 | 08 | 08 | 07 | 07 | 07 | 07 | 06 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
18 | 18 | 18 | 17 | 16 | 15 | 14 | 14 | 13 | 12 | 11 | 11 | 10 | 09 | 09 | 08 | 08 | 08 | 07 | 07 | 07 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
19 | 19 | 19 | 18 | 17 | 16 | 15 | 15 | 14 | 13 | 12 | 12 | 11 | 10 | 10 | 09 | 09 | 08 | 08 | 08 | 08 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
20 | 20 | 20 | 19 | 18 | 17 | 16 | 16 | 15 | 14 | 13 | 12 | 12 | 11 | 10 | 10 | 10 | 09 | 09 | 09 | 09 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+

and now for the defence...

D E F E N C E
| 01 | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
01 | 01 | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
02 | 01 | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
03 | 01 | 01 | 03 | 04 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
04 | 01 | 01 | 03 | 03 | 04 | 05 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
05 | 01 | 01 | 02 | 03 | 04 | 04 | 05 | 06 | 07 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
06 | 01 | 01 | 03 | 03 | 03 | 04 | 04 | 05 | 06 | 07 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
A 07 | 01 | 01 | 02 | 03 | 03 | 04 | 04 | 05 | 05 | 06 | 07 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
T 08 | 01 | 01 | 03 | 03 | 03 | 03 | 04 | 04 | 05 | 05 | 06 | 07 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
T 09 | 01 | 01 | 02 | 03 | 03 | 03 | 04 | 04 | 04 | 05 | 05 | 06 | 07 | 07 | 08 | 09 | 10 | 11 | 12 | 13 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
T 10 | 01 | 01 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 05 | 05 | 06 | 06 | 07 | 08 | 08 | 09 | 10 | 11 | 12 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
A 11 | 01 | 01 | 02 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 05 | 05 | 06 | 06 | 07 | 08 | 08 | 09 | 10 | 11 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
C 12 | 01 | 01 | 03 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 05 | 05 | 06 | 06 | 07 | 08 | 08 | 09 | 10 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
K 13 | 01 | 01 | 02 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 05 | 05 | 05 | 06 | 06 | 07 | 08 | 08 | 09 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
14 | 01 | 01 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 05 | 05 | 06 | 06 | 07 | 07 | 08 | 08 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
15 | 01 | 01 | 02 | 03 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 04 | 05 | 05 | 06 | 06 | 07 | 07 | 08 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
16 | 01 | 01 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 05 | 05 | 05 | 06 | 06 | 07 | 07 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
17 | 01 | 01 | 02 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 04 | 05 | 05 | 05 | 06 | 07 | 07 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
18 | 01 | 01 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 04 | 05 | 05 | 05 | 06 | 06 | 07 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
19 | 01 | 01 | 02 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 04 | 05 | 05 | 06 | 06 | 07 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
20 | 01 | 01 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 03 | 04 | 04 | 04 | 04 | 05 | 05 | 05 | 06 | 07 |
----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+

So what can we learn with the addition of these pieces of data?

Before we start to look more intensively at this new data I must make one point clear. The number given is the number of armies that the attacker (or defender) has at least a 50% chance of having remain. For example, if the attacker attacks with 20 armies against 6, then there is at least a 50% chance that the attacker will have 16 *or more* armies remaining after the attack. Conversely if the attacker were to lose then the defender has at least a 50% chance of having 3 *or more* armies remaining.

But what are all these figures useful for?In risk there are normally two questions that are being asked "How far can I extend this attack" and "Can I hold this position". The best way to answer these questions are of course with an example. So let us look at some crude ASCII art of Africa, as it is depicted on a Risk board.

+-----+-----+ NA - North Africa 5 Armies
|NA |E | E - Egypt 5 Armies
<--| | 5 | C - Congo 1 Army
to | 5 | | EA - East Africa 1 Army
B | +-----| SA - South Africa 1 Army
r +-----+EA | M - Madagascar 1 Army
a |C | 1 +--+
z | 1 | |M |
i +-----+-----+ |
l |SA | 1|
| 1 +--+
+-----------+

Let us create the scenario where you are attacking Africa, as it stands above, from Brazil with 20 armies placed on Brazil. Is it possible to take all of Africa? We will use the tables above to analyze each step.

Step 1: Brazil into North Africa

Our tables tell us that 20 armies against 5 has a probability of winning of 1 (in acutallity its more like 0.999999... but rounding errors take their toll). Our table also tell us that we are likely to have 17 armies left. So we attack and we move all our armies (barring the one that must remain in Brazil) in to North Africa.

At the end of the 1st step we now have 16 armies in North Africa.

Step 2: North Africa into the Congo

15 vs. 1. Probability of winning: 1. Likely number of armies left: 15.

At the end of the 2nd step we have 14 armies in the Congo

Step 3: Congo into South Africa

14 vs. 1. Probability of winning: 1. Likely number of armies left: 14.

At the end of the 3rd step we have 13 armies in South Africa

Step 4: South Africa into Madagascar

13 vs. 1. Probability of winning: 1. Likely number of armies left: 13.

At the end of the 4th step we have 12 armies in Madagascar

Step 5: Madagascar into East Africa

12 vs. 1. Probability of winning: 1. Likely number of armies left: 12.

At the end of the 5th step we have 11 armies in East Africa

Step 6: East Africa into Egypt

11 vs. 5. Probability of winning: 0.950. Likely number of armies left: 8.

At the end of the 6th and last step we have 7 armies in the Egypt

So at the end of this we can see that the probability of taking the African continent as depicted above with 20 armies is `(1 x 1 x 1 x 1 x 1 x 0.950) = 0.950` and you're likely to have 7 armies in Egypt at the end of that.

There is, however, a large flaw with the above calculation - it assumes that at each stage we will still have the most likely number of armies left. Any change in that number will have a large impact on all the following stages. It still does show that twenty armies will have a very good chance of conquering the Africa shown above.

Now let us move on to our final point about defence. Let us take once more our situation above. We shall assume that, after the attack, the attacker then moved (the fortification move) three armies from Egypt to North Africa, so there are now four armies in North Africa and four armies in Egypt. What will this defend us against? Handily this question has already been answered for us. Above we discussed the idea of a "good" chance at winning an attack and we noted that (if we go by 0.66 as a good chance of winning) the attacker must have two more armies than the defender in order to have a good chance. So we can easily see that four armies will defend us against up to six attacking armies against each location

I feel here that I have covered most of the basics of how the statistics of the die effects our strategies in Risk, but if you have any more questions, or if you would like me to look more closely at certain scenarios then please msg me. If there is sufficient demand I will happily put the tables up on the Internet in a prettier format.