Sunday, May 06, 2012

Another puzzle

UPDATE, 4:45pm Sunday:  Oops!  I had to add another condition to make the puzzle work.  See below.

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(Note: This time, the puzzle has some sports content.)

The rules in the Oversimplified Basketball Association (OBA) are as follows: Each team takes turns getting a possession of the ball.  If they score a field goal, they get two points.  There are no three-point shots, rebounds, or fouls.  If there's a turnover or a missed shot, the referee blows the whistle and the other team inbounds the ball to begin their possession.

So, all possessions are independent and identical.  The game lasts exactly 100 possessions for each team.  (Except that if the score is tied, each team takes one extra possession to try to break the tie.  If the score is still tied after that, each team gets one more, and so on.)

The average field goal percentage in the OBA is .500, so the average team scores 100 points.  But some teams have a field goal talent of, say, .550, and score an average 110 points a game, while others have a talent of, say, .400, and score only 80 points a game.

One year, the owners decide the fans like blowouts more than close games.  They suggest changing the rules slightly.

Under the new proposal, if a team makes a field goal, it gets to keep the ball for another possession.  Only when they miss does the ball get turned over to the other team.  The game ends when both teams have an equal number of misses, and they both have passed 100 possessions.  (If the game is tied at that point, the game continues, until each team has missed once more.  If it's still tied, repeat.)

It might help to think of the new rules by a baseball analogy.  The game begins.  The first team gets the top of an "inning", consisting of as many baskets as it can make until it makes an "out" by missing.  Then the second team gets the bottom of the inning.

Innings repeat until, at the end of a given inning, both teams have had 100 possessions ("at bats").  Then the game is over, and the most baskets ("runs") wins.  If there's a tie, play an extra "inning" until the tie is broken.

The owners argue to the fans that this should work.  They reason,

"Suppose team A has a .750 field goal percentage, and team B is .500.  Before the rule change, team A would go 3-for-4, and team B would go 2-for-4.  So, team A would outscore team B by a 3:2 ratio.

"But with the new rules, team A scores 3 field goals for every miss.  Team B scores only 1 field goal for every miss.  So, now, team A should outscore team B by 3:1, not 3:2.

"So, instead of winning by an average 150-100, team A should now win by an average 300-100."

The owners also realize they can sell more commercials, because the games will be longer than before.  For instance, by the time team B gets to 100 possessions to end the game, team A will probably have had 200.

So, the OBA implements the rule change, and there are lots of blowouts, exactly as expected.

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(UPDATE) One more condition, to make the puzzle work: both teams always miss on their 100th possession.

Given that condition, prove that the rule change does not affect the odds of who wins the game.

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Hint: As in the previous puzzle, the solution does not require any fancy math.

At Sunday, May 06, 2012 3:53:00 PM,  Willy Hu said...

By "odds of winning the game", are you talking about the ratio of points (in your example, the change from 3:2 to 3:1)?

At Sunday, May 06, 2012 3:55:00 PM,  Phil Birnbaum said...

I mean, having scored more points than the opponent at the end of the game.

You win if you have more points regardless of the ratio.

At Sunday, May 06, 2012 3:57:00 PM,  Phil Birnbaum said...

That is, show that team X has the same probability of winning (scoring more points) regardless of which of the two sets of rules is used.

At Sunday, May 06, 2012 4:11:00 PM,  Phil Birnbaum said...

Sorry, I think I screwed up. The question doesn't work. I'll delete the post eventually if I can't fix it.

At Sunday, May 06, 2012 4:48:00 PM,  Phil Birnbaum said...

OK, I added one more condition, that both teams always miss on their 100th possession. Now it works, I think.

If anyone wants to check my solution before I post it, send me an e-mail. My last name (at sign) sympatico.ca .