"What are the odds?" Bar Games Edition

Received two bar games-related questions in the past few days - both fitting "what are the odds?" style probability inquiries.

Yahtzee $5 Dice Roll

First: the game is a dice roll, with the aim to roll 5 dice to get a Yahtzee (all 5 dice match). You get 5 attempts, but unlike Yahtzee, you don't get to keep any matching dice - you have to reset your roll each time. Entry fee is $5, and the winner gets the pot of all entry fees (minimum pot size is $50). What does the pot size have to be for you to want to enter?

What we're looking for is at what point the expected value (E[V]) is positive. The odds of rolling a Yahtzee on one roll is if any one number matches on 5 dice, with 6 possible number matches. Thus (1/6) ^ 5 = 1 / 7776, times 6 = 6 / 7776 = 1 / 1296, or 0.08%. Verified here. Thus the odds of NOT winning on one roll is 1 - 0.08% = 99.92%.

Since there is no rollover, the math is pretty straight forward. The odds you LOSE 5 times is 99.92% ^ 5 = 99.6%. So the odds you WIN is 1 - 99.6% = 0.4%.

Your expected loss is 99.6% * -$5 = -$4.98, so we want to calculate at what point your expected winnings would be more than $4.98, and thus create a breakeven point. This can be calculated by your expected loss / odds you win, so $4.98 / 0.4% = $1,293. This is how large the pot needs to be before you'd be indifferent between lighting $5 on fire and playing the dice game; way higher than the $50 minimum pot size.

Odds You Miss Every Single Trivia Question

Second: the game is bar trivia, with 20 questions with 4 multiple choice answers each. Say you're trying to tank the game and miss every single question (for draft position next round maybe??) and you christmas-tree each question at random. What are the odds you get at least one question correct by accident?

If each guess is truly random, you have a 75% chance of getting a single question wrong. So the odds you get EVERY question wrong are 75% ^ 20 = 0.3%, meaning that it is extremely likely you will get at least one question right, at 1 - 0.3% = 99.7%.

There is actually a better chance you win the dice game than miss every trivia question!