Goblin Menace

Probability Level 1

You are playing Dungeons and Dragons with five of your friends. You are a first-level wizard and you and your friends are fighting a Goblin Boss (17 AC, 21 HP). One of your friends, a first-level fighter, dealt six damage against the Goblin Boss on his turn. It is now your turn and you decide to cast magic missile against the wounded Goblin Boss. What is the probability that you will kill the Goblin Boss on your turn (i.e: the goblin boss is dead by the end of your attack)?

Important Details:

  • Your spell attack modifier is +5.

  • If you roll a 20 on your attack roll, your damage is doubled.

  • Magic Missile shoots three glowing darts of magical force and each dart deals 1d4 + 1 damage against a target if it hits the target. You direct all three darts against the Goblin Boss.

  • Assume that you will not be moving or taking a bonus action.

  • When the Goblin Boss's HP is less than or equal to zero HP, they die.

Relevant rules of Dungeons and Dragons:

AC = Armor Class. AC is a measure of the defenses of a creature and is instrumental in calculating if an attack roll succeeds or fails.

HP = Hit Points. HP is a measure of the life of a creature. If damage is dealt against the creature, the damage is subtracted from the creature's HP.

dN, where N is an integer greater than four = N-sided die. For example, a d20 is a 20-sided die and a d4 is a 4-sided die.

Magic missile is a spell attack, which has the following rules:

  1. Roll a d20 and add your spell attack modifier to the outcome of the roll.

  2. Compare the total to the target creature's AC. If the total is equal to or larger than the creature's AC, the attack hits, otherwise, the attack misses and no damage is dealt. In the case where the total equals the creature's AC, the damage is halved.

  3. To determine the damage of the attack, roll the appropriate number and type of dice and add the relevant modifiers. For magic missile, this would be 1d4 + 1 damage for each dart.

  4. Subtract the damage dealt from the target creature's HP.

An example of a spell attack using magic missile is demonstrated below:

  1. I roll a d20 and get a 9. I add my spell attack modifier (+6) for a total of 15.

  2. Comparing this to the target creature's AC (15), I find that my attack hits and does half damage.

  3. I direct all of my magic missiles at the target creature. I roll 3d4 and add 3, which yields 12. The total damage is then halved and becomes 6.

  4. I subtract 6 from the target creature's HP (10). The creature's new HP is 4.

At most 4% Equal to 5% Between 5% and 6% Between 4% and 5% At least 6%

Alexander McDowell by Alexander McDowell , 18, USA

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1 solution

Keep in mind that the HP of the Goblin Boss is 15, not 21. We can now solve this problem using cases.

Case 1 : The attack roll is less than the Goblin Boss's AC:

Call this probability p 1 p_{1} . If the attack roll is less than the Goblin's AC, then no damage is dealt. Thus, p 1 = 0 p_{1} = 0 .

Case 2 : The attack roll is equal to the Goblin Boss's AC:

Call this probability p 2 p_{2} . If the attack roll is equal to the Goblin's AC, then the damage is halved. Since the max damage of magic missile is 15 (assuming the damage is not doubled due to rolling a 20), half of 15 is 7.5. 7.5 damage is not enough to kill the Goblin Boss, so p 2 = 0 p_{2} = 0 .

Case 3 : The attack roll is greater than the Goblin Boss's AC (but is not 20):

Call this probability p 3 p_{3} . In this case, we are able to damage the Goblin. Subtracting the spell attack modifier from the Goblin Boss's AC, we find that we have to roll either a 13, 14, 15, 16, 17, 18, or 19 for our attack to hit. In addition, to kill the Goblin Boss, we must deal maximum damage (again, assuming the damage is not doubled due to rolling a 20). Thus, p 1 = 7 20 ∗ ( 1 4 ) 3 = 7 1280 p_{1} = \frac{7}{20} * (\frac{1}{4})^{3} = \frac{7}{1280} .

Case 4 : The attack roll is 20:

Call this probability p 4 p_{4} . Since the damage is doubled, the sum of our 3d4 + 3 must be at least 8. We need to then find the probability of four dice having a sum of at least 5. This is equivalent to the complement of four dice having a sum of at most 4 (i.e: P ( s u m ≥ 5 ) = 1 − P ( s u m ≤ 4 ) P(sum \geq 5) = 1 - P(sum \leq 4) . P ( s u m ≤ 4 ) P(sum \leq 4) can happen for dice sums of 3 or 4. Three four-sided dice can only sum to 3 if I roll a one on all of the dice, yielding one possibility. Three four-sided dice can only sum to 4 if I roll a one on two dice and I roll a two on one dice, yielding 3 ! ( 1 ! ) ( 2 ! ) = 3 \frac{3!}{(1!)(2!)} = 3 possibilities. Thus, P ( s u m ≥ 5 ) = 1 − ( 3 + 1 64 ) = 15 16 P(sum \geq 5) = 1 - (\frac{3 + 1}{64}) = \frac{15}{16} . Thus, p 4 = ( 1 20 ) ( 15 16 ) = 3 64 p_{4} = (\frac{1}{20})(\frac{15}{16}) = \frac{3}{64}

Thus, the total probability of defeating the Goblin Boss on your turn is p t o t a l = p 1 + p 2 + p 3 + p 4 = 0 + 0 + 7 1280 + 3 64 = 67 1280 p_{total} = p_{1} + p_{2} + p_{3} + p_{4} = 0 + 0 + \frac{7}{1280} + \frac{3}{64} = \frac{67}{1280} which is slightly larger than 5%.

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