Puzzle: How to make the dice game fair

Reply #3

Wavepack — Tue, 07 Feb 2023 20:27:17 GMT

This type of Bayesean conditional reasoning can be used to solve the probability of a winning game of tennis given p=probability of winning a point in tennis. You will find a large slope at p=50% of the P(Win Game) vs P(Win Point) plot. ie, small point win advantage turns into a large probability of game win. You can compute the exact value of this slope.

Also, you can derive why a higher speed 1st serves make sense to maximize p, despite greater fault rates of higher speed serves.

Reply #2

db101 — Mon, 06 Feb 2023 20:26:27 GMT

I like your algebra solution. What made this one hard and not intuitive, for me at least, is that you know the prob of rolling snake eyes in 1 roll is 1/36, and the prob of rolling 7s back to back in 2 rolls is also 1/36, so it feels like it should be fair. But you know it can't be because having to do something twice in a row is harder

Reply #1

cottoneyedjoe — Mon, 06 Feb 2023 04:24:01 GMT

The game is still not fair, because now Alice has only a 6/13 ( 46.15%) chance of winning. Brad's likelihood of winning is 7/13 ( 53.85%).

Here's how I solved it. Suppose Brad's probability of winning is Z. On any given roll there are three outcomes:

Sum = 2, with probability 1/36 Sum = 7, with probability 6/36 Sum = something other than 2 or 7, with probability 29/36

If the first roll is a sum of 2 (first case), he wins automatically.

If the first roll is a sum of 7 (second cas... [ More ]

Puzzle: How to make the dice game fair

db101 — Sat, 04 Feb 2023 02:40:39 GMT

Alice has a pair of standard six-sided dice. She says to her little brother Brad, I'm gonna keep rolling these dice until I get either a sum of 7 or 2. If the sum is 7, I win. If the sum is 2, you win. Whaddya say Brad?

Brad says, No way, cheater. Getting sum of 7 is six times more likely than getting a sum of 2! How about you keep rolling until you get either a sum of 7 TWICE IN A ROW, or a sum of 2. If you roll a sum of 2, I win. If you roll a sum of 7 twice in a row, you win.

Alice... [ More ]