• when one player is attacking another player’s territory, the maximum
troops they can attack with is 3, so they roll 1, 2, or 3 dice,
depending on how many troops they have
• the defender can only use a maximum of 2 troops, so they roll 1 or 2
dice
• The player with the highest roll of dice, is the winner of that battle,
so if the attacker wins he gains control of the territory, and if the
defender wins he “kills” one or two of the attackers troops
• if there is a tie, the defender wins
• the attacker has to always leave at least 1 troop in the territory he/she
is attacking from
Situation
If Giorgio is attacking Papa Teo with 10 troops against 1 and
Giorgio will continuously attack until he uses all of his troops,
what is the probability that Papa Teo will stay alive?
P3a = probability of dad winning when Giorgio attacks with
maximum of three troops against one
P2a = probability of dad winning when Giorgio has three troops
left, therefore having to attack with two troops against one
P1a = probability of dad winning when Giorgio has two troops
left, therefore having to attack with one troop against one
P3a = 1/6 +1/6*(5/6)3+1/6*(4/6)3+1/6*(3/6)3+1/6*(2/6)3+1/6*(1/6) 3
P2a = 1/6 +1/6*(5/6)2+1/6*(4/6)2+1/6*(3/6)2+1/6*(2/6)3+1/6*(1/6)2
P1a = 1/6 +1/6*(5/6)+1/6*(4/6)+1/6*(3/6)+1/6*(2/6)*(1/6)
1/6 +1/6*(5/6)3+1/6*(4/6)3+1/6*(3/6)3+1/6*(2/6)3+1/6*(1/6) 3
Probability 1/6 is the probability
of dad
of papaTeo getting 5,
winning and *(5/6)3 is the
with a six probability of Gio
not getting 6 with 3
dice
1/6 is the
probability of
papaTeo getting
4, and *(4/6)3 is
the probability
of Gio not
getting 5 or 6
with 3 dice
1/6 is the probability
of papaTeo getting 3,
and *(3/6)3 is the
probability of Gio not
getting 4 5 or 6 with 3
dice
P3a = 34 %
P2a = 42.1%
P1a = 58.333… %
1/6 is the
probability of
papaTeo getting 2,
and *(2/6)3 is the
probability of Gio
not getting 3 4 5 or
6 with 3 dice
1/6 is the probability
of papaTeo getting
1, and *(1/6)3 is the
probability of Gio
not getting 2 3 4 5 or
6 with 3 dice
P3a happens 7 times in this situation because Giorgio
can attack with 3 troops every battle until he has 3 out
of the ten left. P2a happens only once, because Giorgio
can attack with a max of two troops for one battle when
he starts with 3 troops. P1a occurs just once, when
Giorgio loses all but 2 troops, and can only attack with
one
Probability = (P3a)^7 * P2a * P1a
P= (.34)^7 * .581 * .583
P= 0.01%
survival probability vs attack troops
Number of attack troops
Defender's % of victory
9
0.01%
8
0.04%
7
0.11%
6
0.33%
5
0.97%
4
2.85%
3
8.36%
2
24.58%
1
58.33%
Papi
Gio
Nic
Paolo
Mamma
=
=
=
=
=
5%
0.3%
0.7%
0.6%
0.05%
Mom Wins!!!
0.05% is the closest to 0.01%
So if Giorgio has 10 troops and is
attacking Papa Teo who has only
one troop, Papa Teo has a 0.01%
chance of survival. Papi, I hope
you have better luck next time,
unless I’m attacking you.