Q1 I have been back at playing 'Master of Magic' again. You may recall that this is a video game based where I play a wizard trying to take over the world.

This time Tony the Barbarian is facing a group of 3 distinct Skeletal Archer units. They are 10 squares away, so he has to spend 10 full turns moving towards them before he will be within range of the three units and can attack them with his Ethereal Axe of Vanquishing. Each Skeletal Archer unit is capable of firing up to 8 volleys of arrows, at which point they will run out of ammunition and be unable to attack, though they can still defend themselves with their shields.

When a Skeletal Archer unit attacks Tony, there is an 60% chance they will miss, and if they do hit him, there is a 30% chance that he can block it with his shield, taking no damage. If he is hit and does not successfully block, he will suffer one hit point of damage per volley of arrows. He only has 5 hit points (HP), and will lose the battle and be forced to retreat if his HP reaches 0. That could be a problem, particularly since, while he is moving towards them, he cannot attack. But that is what you get when you bring an axe to an archery fight.

Once the skeletons are out of ammunition, Tony still has to use his magic axe to defeat them. Each turn, he can make one attack. To destroy a skeleton, he first has to hit it (i.e. roll 16 or better on a 20-sided dice) and then the skeleton has to fail its saving throw (roll below a 14 on a 20-sided dice).

At the start of combat, Tony is pondering all this while staring at the Skeletal Archers, there are 50 moves available before the battle ends and 'All units retreat exhausted.'

a. What is the probability that Tony emerges victorious?

b. What is the probability that Tony emerges victorious AND is unhurt by the skeletons?

c. What is the probability that the Skeletons win and Tony has to retreat?