This is probably not new to any student of statistics - yet I can't find the name of that "paradox" to search for.

Imagine the richest person on earth holds a game:

A participant is invited to roll a 10-sided die. If he rolls a 10, he gets executed. For a roll of 1 through 9, he wins a million dollars, and the game is repeated with ten times the number a participants. For each round of the game, the die is rolled once. (E.g. the game host rolls instead of the participants.)

This continues until the first 10 is rolled. Then the game stops, all current participants are executed, and the game never repeats. For example if the game ends in the 4th round, 1+10+100=111 players are rich while 1000 are dead.

Your relatives learn the rules, and learn that you participated. They know that:

  • You have a 90% chance of being a millionaire.
  • 90 % of all total participants get executed.

So what is it? Are you most likely rich or most likely dead? How can both answers be valid?

  • $\begingroup$ "Game of Life" is not the best name, as it is already used en.wikipedia.org/wiki/Conway%27s_Game_of_Life $\endgroup$ – Tim Nov 6 '19 at 19:45
  • $\begingroup$ It does not appear that a winner moves into the next round; given that, it should be clear that you have a 90% chance of being a millionaire and a 10% chance of being dead, since for you personally it's just a one-round game. If the winner moves into the next round, and keeps moving, eventually (with probability 1) a 10 will be rolled and the multimillionaire will be executed, leading to a 100% chance of being dead (and a millionaire.) $\endgroup$ – jbowman Nov 6 '19 at 20:40
  • $\begingroup$ @jbowman: No one moves between games. You participate in exactly one round. After that you either are rich or dead. The paradox comes from the fact while you have a 90% chance of winning your game, since the number of participants rise between games, the total number of loosers outweight the total number of winners. $\endgroup$ – Zsolt Szilagy Nov 7 '19 at 14:42

If you participate, you have a 90% chance of becoming a millionaire and only a 10% chance of being executed. The odd nature of the game leads to the fact that, in each instance, slightly over 90% of participants will be executed.

However, even this is misleading depending on how participants are chosen. If we assume that millions of people sign up and are selected for participation at random, most volunteers will never participate at all. So, if you volunteer, your most likely result is actually to not participate at all.

I get where you are going with calling this a paradox, but it is conflating two rather different statements.

  • If you participate, there is a 90% chance that you will become a millionaire
  • Approximately 90% of participants are executed each time the game is played

Both are true, and these statements are not in conflict with each other. The primary reason is that the participation is for a single round and the probability of rolling a 10 is no higher (or lower) depending on the number of participants. However, when we know how many volunteers were selected to participate, the results are fixed (if 11 participated, a ten was rolled in the second round; if 111 participated, a ten was rolled in the third round; etc.).

Here is my previous answer based on the previous description of the game.

I believe that you are misstating the probabilities, and there is some ambiguity in the rules. The way they are phrased here, all participants in a given round are executed if any participant rolls a 10.

If this is the case, the proababilities are very different for each round:

  • In round one, you have a 10% chance of execution and 90% chance of becoming rich
  • In round two, you have 65% chance of execution and 35% chance of becoming rich
    • This is the probability of rolling no tens in 10 rolls
  • In round three, you have 99.99973% chance of execution and 0.0023% chance of becoming rich
    • This is the probability of rolling no tens in 100 rolls

The question of your overall probability of hitting it rich vs being executed depends on your ability to control when you enter the game. If you simply sign up to participate at a randomly selected stage, you are far more likely to be entered into a later round. Here, for example (ignoring round 4 because the game will reach that round less than 1 in 10,000 times), you have:

  • 1% chance of playing in round one
    • so 0.1% chance of dying, 0.9% chance of being rich);
  • a 9% chance of playing in round two
    • 2.8% chance of getting rich; 5.27% chance of dying;
    • This is after removing the 10% chance that the game never reaches this stage; and
  • a 90% chance of playing in round three
    • 68.6% chance of the game not reaching this round
    • otherwise you are almost certainly going to die

So, your overall probabilities are:

  • 62.6% chance of not playing at all
  • 33.6% chance of dying
  • 3.7% chance of getting rich

(I rounded more than I should have, but it gets us close enough.)

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  • $\begingroup$ "The way they are phrased here, all participants in a given round are executed if any participant rolls a 10." Sorry, that ambiguity is my fault - edited to clarify that the dice is rolled once per round. $\endgroup$ – Zsolt Szilagy Nov 6 '19 at 18:13

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