Scrapball
Scrapball is a children's game in which player's attempt to hit one another with dodgeballs. If you are hit, you must sit down. If the person who made you sit down you gets hit, you may stand (along with everyone else the hit person made sit down). A winner is declared when they are the last person standing.
Assuming there are 4-players of equal skill (i.e. equally likely to hit someone or get hit), how many hits will happen on average before the game is over?
The answer is 7.
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1 solution
Model the problem using partitions of the number of players. If there are 6 players the game starts as 1,1,1,1,1,1, and if you hit someone they are added to your "group" and your number would iterate up by 1. A partition might look like 3,2,1 which would mean 1 player has hit 2 players 1 player has hit 1 and 1 has not hit anyone. If 1 hits 3, the new partition is then (try to figure it out)
1,1,2,2 (Ask yourself why)
Alright so now we can analyze a 4 player game. We make a flowchart with starting with 1,1,1,1 which displays the probability of moving from one state to another. Here is an image for the 4 player flowchart with associated values (try and draw it for yourself before you look).
From here we ignore the first state since we can never return to it and just remind ourselves to add 1 to our solution at the end. We anchor ourselves at (2,1,1) and calculate that bubble's expected hits until it reaches 4, which we'll call E.
E = 6 1 ( 2 ) + 6 1 ( 3 ) + 3 1 ( E + 1 ) + 6 1 ( E + 2 ) + 6 1 ( E + 3 )
There is 1/6 chance we make it to 4 in 2 hits, and 1/6 we make it in 3. There is a 1/3 chance we return to E after 1 hit and 1/6 we return after 2 or 3 hits.
Solving amazingly gives us an integer, E = 6. Reminding ourselves to add 1 for the initial hit brings us to 7.
If you want a real challenge, try the same analysis for 5 players.