How To Figure Out The Average Number Of Questions An HQ Player Answers Correctly

If you’re not familiar with HQ, it works like this: during each game, a number of people will compete. For the sake of simplicity in this thought experiment, let’s say it’s a million people.

Players are asked 12 (sometimes more, but let’s not quibble) questions, each with three multiple-choice possible answers. After each question, you’re told whether or not you got the answer right, plus how many people supplied each answer. Once you answer incorrectly you are eliminated.

So let’s say a hypothetical game looks like this.

Question 1:

Right answer: 975k players.
Wrong answer a: 12,500 players.
Wrong answer b: 12,500 players.

975k players remain playing.

Question 2:

Right answer: 800k players.
Wrong answer a: 100,000 players.
Wrong answer b: 75,000 players.

800k players remain playing.

Question 3:

Right answer: 500k players.
Wrong answer a: 200k players.
Wrong answer b: 100k players.

Question 11 (begins with 4,650 players still playing):

Right answer: 1,650 players.
Wrong answer a: 2,000 players.
Wrong answer b: 1,000 players.

Question 12:

Right answer: 375 players
Wrong answer a: 1,000 players
Wrong answer b: 275 players

Given this information, how can we crunch the numbers to say the average number of questions a player got right in this game?

You have to be careful about what question you’re asking. If you want the average number of questions a player got right, then you add up the total number of right answers, and divide by the total number of players. But that won’t tell you anything about the number of people who would have gotten question 12 right, if they hadn’t already been eliminated from the game. That might be more complicated, depending on things you haven’t told us.