2. 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?

3. P 3a = probability of dad winning when Giorgio attacks with maximum of three troops against one P 2a = probability of dad winning when Giorgio has three troops left, therefore having to attack with two troops against one P 1a = probability of dad winning when Giorgio has two troops left, therefore having to attack with one troop against one

4. P 3a = 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 P 2a = 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 P 1a = 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 1/6 is the probability of papaTeo getting 5, and *(5/6) 3 is the probability of Gio not getting 6 with 3 dice Probability of dad winning with a six 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 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 P 3a = 34 % P 2a = 42.1% P 1a = 58.333… %

5. P 3a happens 7 times in this situation because Giorgio can attack with 3 troops every battle until he has 3 out of the ten left. P 2a happens only once, because Giorgio can attack with a max of two troops for one battle when he starts with 3 troops. P 1a occurs just once, when Giorgio loses all but 2 troops, and can only attack with one Probability = (P 3a )^7 * P 2a * P 1a P= (.34)^7 *.581 *.583 P= 0.01%

6. Number of attack troopsDefender's % of victory 90.01% 80.04% 70.11% 60.33% 50.97% 42.85% 38.36% 224.58% 158.33% survival probability vs attack troops

7. Papi = 5% Gio = 0.3% Nic = 0.7% Paolo = 0.6% Mamma = 0.05% Mom Wins!!! 0.05% is the closest to 0.01%

8. 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.