# Expected Earnings in HQ Trivia

9/5/18

HQ Trivia is a popular live trivia game show. The game has 12 questions (although the first 3 are very easy) with 3 choices for each question. If you get a question wrong your game is over and you get \$0. But if you get all 12 questions correct, you win the jackpot, maybe \$5,000 or \$10,000! Well, not exactly... you win the jackpot/the number of players who also won. For example, if the jackpot is \$5,000 and 700 people win, then each winner wins \$5,000/700 = \$7.14.

After playing HQ Trivia a few times, I wanted to approximate via statistics and simulation how much one could reasonably expect to win. Consider modeling the game as a binomial distribution with n = 9 (and not n = 12 because the first 3 questions are simple - "gimmes"), and two scenarios. The first scenario is random guessing, where the probability of getting each question correct is p = 1/3. The second scenario considers skill, where p is larger than 1/3. In each scenario, we have

Prob(r questions correct out of 9) = nCr(9,r)*pr*(1-p)9-r

where nCr is the combination function, and is nCr(n,r) = n!/(r!(n-r)!), where "!" denotes the factorial function.

## Random Guessing Scenario

 Questions Correct (r) Probability of getting r questions correct 0 .026 1 .117 2 .234 3 .273 4 .205 5 .102 6 .034 7 .007 8 .001 9 .0000508 (or about 1 in 19,683)

The expected number of questions you can expect to get correct is 9*(1/3) = 3.

Let's now consider the case of skill. To do this, I did the same calculations above, but I let p = 1/2. One thing to note, is that the more skilled people are, the more winners there will be, and winners will therefore receive less than if there is less skill present.

## Skill Scenario

 Questions Correct (r) Probability of getting r questions correct 0 .002 1 .018 2 .070 3 .164 4 .246 5 .246 6 .164 7 .070 8 .018 9 .002 (or about 1 in 512)

The expected number of questions you can expect to get correct is 9*(1/2) = 4.5, so 4 or 5.

There are some other variables to consider. First, there is the number of people, N, playing the game at the start. This number decreases as the game continues and more and more people get questions wrong. The second variable is the prize amount, T.

Your expected earnings, if you make it through to the end are therefore

T/(N*p9) dollars

Substituting in some other numbers, say N = 300,000, T = \$5,000, and people are skilled with p = .75. The winners may expect to win about \$.22 each. See this spreadsheet to try out some other values. I believe that you are probably likely to win an amount in the range \$0 to \$10 for reasonable values of p, N, and T.

Most people probably do not play for financial reasons, but for reasons of having fun, learning trivia, and social interaction. Nothing wrong with that!