I'm trying to calculate what my odds are as a level 20 Champion to score at least one critical hit on my turn, and how many I can expect.
I make four attacks per attack action I take, and can Action Surge for another four, bringing my total to 8. I score a critical hit on a roll of 18-20 on the d20, and if I get even one, my Great Weapon Master feat lets me make a fifth (or in this case ninth) attack.
I'd also be curious to know what my odds are if I'm rolling with advantage.
Best Answer
Probability of at least one crit
The Great Weapon Master feat allows making an extra attack when scoring a critical or reducing a creature to zero hit points. I'll not cover the latter case since its probability is very situational, and the former has no relevance here since the extra attack only happens if one already scored a critical hit. The easy way to approach this question is to consider the inverse case: what is the probability that you score no critical hits on eight attacks?
Each attack scores a critical hit on a 18, 19 or 20, so the probability of not critting on a given attack is \$\frac{17}{20}\$. The probability of eight attacks with no critical hits is therefore \$(\frac{17}{20})^8\$, roughly \$0.27\$. This means you score at least one critical hit with a probability of \$1- (\frac{17}{20})^8\$, or roughly \$0.73\$.
With advantage
With advantage the probability of each individual blow critting is \$2 \times \frac{3}{20} - (\frac{3}{20})^2 = \frac{120}{400} - \frac{9}{400}\ = \frac{111}{400} ≃ 0.28\$. Therefore, the probability of at least one critical hit on eight attacks with advantage is, using the same reasoning as above, \$1 - (\frac{289}{400})^8 ≃ 0.93\$.
Expected number of criticals
Expected values are generally quite easy to work with, because they work in a linear fashion: the expected critical hits per a single attack is \$\frac{3}{20}\$, so the expected critical hits per eight attacks is \$\frac{3}{20} \times 8 = \frac{24}{20}\$. You will on average score \$1.2\$ criticals per round assuming you make all eight attacks, but this is before we factor in the extra attack from Great Weapon Master.
The extra attack given by the Great Weapon Master triggers with the probability \$1- (\frac{17}{20})^8\$ we calculated earlier. Since it uses one's bonus action, it will trigger only once per turn at most. When it triggers, it adds an expected \$\frac{3}{20}\$ critical to the total number of crits per turn. Taking into account the probability of triggering, the expected number of critical hits added is \$(1- (\frac{17}{20})^8) \times \frac{3}{20} ≃ 0.11\$. Therefore the final expected number of crits per round where one makes all eight attacks is roughly \$1.31\$.
With advantage
Each attack has a probability of \$\frac{111}{400}\$, so eight attacks yields an expected value of \$\frac{111}{400} \times 8 = 2.22\$ critical hits before accounting for the GWM extra attack. Accounting the extra attack, we add \$(1 - (\frac{289}{400})^8) \times \frac{111}{400} ≃ 0.26\$ giving us a total of 2.48 criticals expected per round.