Pizza Distribution Puzzle

Source: xkcd wiki

Problem:
The king of the universe has decided to play a game. To start, he selects 1 person. He then flips two fair coins - if they both come up heads, the person gets a free pizza and the game is over. For any other result, he sends the person home and selects 2 new people, where he does the same 2-coin flip to decide if they each get a pizza. If they don’t, he picks 4 people at random, then 8, and so on, doubling each round. If you are selected but don’t win, you can’t be selected again – and you can assume the population is extremely large so there’s no chance of running out of contestants.

You are sitting at home when you get a call – you have been selected to play the game. What is the chance that you will get a free pizza?

You don't know which round number it is, but if you ask, the king will tell you. Does it matter?

 Disclaimer: Very easy problem!

Update (31 January 2013):
Title changed to "Pizza Distribution Puzzle" from "Pizza Paradox Puzzle" - as pointed out by Vivek Ranjan Nema in comments, there is no paradox. My mistake. Apologies.
Solution posted by Rushabh Sheth, Pratyush Rathore, Nathan Jacobson, Marjansek, Shyam Raj in comments!

Comments

  1. rushabh sheth11:23 AM, December 29, 2012

    answer 1/4
    matters not which round.
    each round has same probability of winning.
    we are not interested in what round is going on, just what our probability of winning is.

    ReplyDelete
  2. Pratyush Rathore1:24 PM, December 29, 2012

    1/4.
    My fate of pizza is decided by a single event, the coin toss.

    Everything prior to this, determines the chance whether or not, I get to play the game, which is the starting point in this case.

    ReplyDelete
  3. Unknown6:43 PM, December 29, 2012

    I guess the probability is still 1/2 as I have already been selected for the contest.

    ReplyDelete
  4. Unknown8:00 PM, December 29, 2012

    Isn't it just a 25% chance since if you are getting a call the previous rounds all lost and your outcome is not influenced by the other players in your round

    ReplyDelete
  5. Anonymous8:23 PM, December 29, 2012

    1/4; probabilities are independent.

    ReplyDelete
  6. Unknown11:16 PM, December 29, 2012

    If the first person did not get pizza, the king selects 2 people. I am not sure, does he flip 2 coins for both and both can get free pizza, or does he flip 2 coins for person 1 and if he gets free pizza the second does not?
    Rika

    ReplyDelete
    Replies
    1. Unknown9:48 PM, January 02, 2013

      No matter what is your order in the queue. But for every person, the king flips two coins. And both have to show heads for you to get the pizza. Which means (1/2)*(1/2). Hence 1/4.

      Delete
  7. x2:26 AM, December 31, 2012

    1 - (3/4)^k, where k is round number

    ReplyDelete
  8. Vivek 9:01 AM, January 09, 2013

    What is the paradox?

    ReplyDelete
  9. Simão6:24 AM, June 29, 2015

    the reason for this to be a paradox, is that, in a way, your chance of winning should be 1/4. However, as someone said in the talkpage for this problem, over at the skcd wiki:

    Assume all the coin tosses were done ahead of time - shouldn't matter right? Now choose the people corresponding to each round. More than half these people are in the final round. So if you are called there's more than 50% chance you're a winner.

    Which contradicts the intuitive (and logical) idea tht the probability is 25%. Hence the paradox.

    ReplyDelete
  10. sky2:10 PM, June 03, 2021

    If we are looking back in hindsight - P(I won the game/I had been called during the game) then the probability is more than 50%. In this hindsight view, we have fixed the total games played as finite number. So the problem that this view is considering is this - P(I won the game/I had been called during a game that went on till N plays)

    but if we are looking at present - P(I win the game/I just got called) then it is 1/4. In this view we have not fixed how many tosses in this game will be played or have been played till now. So this case is a more generic consideration of the question - P(I win the game which could go on till any number of tosses/I just got called)

    ReplyDelete
  11. Pratik Poddar1:11 PM, September 08, 2026

    Nicely put - this is exactly the resolution of the apparent paradox. Ex-ante, when you get the call, your chance of pizza is just the chance your two coins both come up heads: 1/4. Ex-post, conditioning on the game having run a fixed number of rounds, the winners are the entire final round, which is as large as all previous rounds put together, so a majority of everyone selected ends up with pizza. Both views are consistent; they answer different conditional questions. (replied using AI)

    ReplyDelete

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