perplexus.info :: Probability : Winning The Regatta

Winning The Regatta (Posted on 2005-11-28)

A regatta is a series of sailboat races. In the fleet where I race, the regatta winner is determined using a method called "Low Point Scoring". In any given race, the 1st place boat gets 1 point, the second place boat gets 2 points, the nth place boat receives n points. Individual races never have ties for any positions. The overall regatta is won by the boat with the lowest total number of points for all races. (If there is a tie for lowest total points, then the regatta is won by whichever of the tied boats had the better performance in the last race).

Consider a relatively small fleet of only 4 boats, each of which is equally likely to win any given race.

a) If there are only two races, the boat that wins the regatta will have a score of 2, 3, 4 or possibly even 5. What is the expected value of the winning score?

b) If there are three races, what is the expected value of the winning score? (I found even this simple case hard to calculate exactly, and I am hoping that somebody will come up with a better method than mine. And yes, I know that it is easy to simulate.)

c) If there is a large number of races, how might I approximate the expected winning score? (Among other things, I think I'd welcome a simulation here)

Submitted by Steve Herman

Rating: 4.5000 (2 votes)

Solution: (Hide)

a) Two races.

A score of 2 will win if the boat who wins the 2nd race also won the first race.

Probability = 6/24.

A score of 5 will win if the boats finish the second race in exact reverse order from the first race.

Probability = 1/24.

A score of 3 will win if the first place boat from the first race finishes 2nd in the second race.

Probability = 6/24.

Or if the 2nd place boat from the first race finishes 1st in the 2nd race.

Probability = 6/24.

But this double-counts the possibility that both things happen (probability) 2/24.

Altogether 10/24.

A score of 4 therefore wins with probability 7/24.

And the expected winning low score = (2*6 + 3*10 + 4*7 + 5*1)/24 = 75/24 = 3 1/8.

b) Three races -- Don't know

c) Many races -- Don't know

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
maybe simpler?	mickey	2006-01-29 21:58:26
re(2): Charlie I wonder...	Steve Herman	2005-12-06 12:28:13
re: Charlie I wonder...	Charlie	2005-12-04 11:19:41
Charlie I wonder...	Dan	2005-12-04 02:05:00
2-7 races exactly; simulation above	Charlie	2005-11-28 10:58:21
4 boats, 2 races or 3 races	goFish	2005-11-28 10:21:18

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