In the game of Bocce, two opponents each throw four balls of their color, trying to get closest to a small neutral ball called the pallino. The player with the ball closest to the pallino scores however many of their balls are closer than the closest of their opponent.
Consider two opponents with zero skill. Although their balls all remain in play, the distances of the eight balls to the pallino are completely random.
Find the winning player's expected number of points.
Extension.
The sport of curling, while played on ice, uses similar scoring except that each team curls eight stones towards a fixed point called the 'tee.' After curling all the stones, the team with the stone closest to the 'tee' scores however many of their stones are closer than the closest of their opponent. Only stones in the 'house' can score.
Again, imagine the teams have zero skill but all of the 16 stones ending up in the house but at a random position.
Find the winning team's expected number of points.
for n=1:10
trials=repmat('ab',n);
trials=trials(1,:);
way=uniqueperms(trials);
ct=0;
for i=1:size(way,1)
tr=way(i,:);
for j=2:n+1
if tr(j)~=tr(1)
break
end
end
ct=ct+j-1;
end
ev=sym(ct/size(way,1));
fprintf('%2d %6s %8d %8d %13f\n',n,ev,ct,i, double(ev))
end
The function uniqueperms is available online in the MATLAB file exchange.
The table produced by the above program shows for n = the number of balls or stones rolled or shot by each person or team, the expected score for the winning team as a rational number, the numerator and denominator of the unreduced fraction and the expected value as a decomal.
n expected unreduced decimal
score num den
1 1 2 2 1.000000
2 4/3 8 6 1.333333
3 3/2 30 20 1.500000
4 8/5 112 70 1.600000
5 5/3 420 252 1.666667
6 12/7 1584 924 1.714286
7 7/4 6006 3432 1.750000
8 16/9 22880 12870 1.777778
9 9/5 87516 48620 1.800000
10 20/11 335920 184756 1.818182
For the case of Bocce (n=4), there are 70 unique orders of distance, in which the total of points scored for all 70 is 112, giving an average score of 8/5.
For curling (n=8) it would be 22,880 points scored over the total of 12,870 possible orders, for an average of 16/9 points scored.
As the number of object pitched per team increases, the expected points approaches 2.
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Posted by Charlie
on 2021-06-02 10:32:07 |
solution
