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.
Given one of your balls is closest to the pallino, there are 7C3 = 35 ways to order the rest of the balls in terms of distance from the pallino. In 6C3 = 20 of them you don't have a ball in 2nd place; in 5C2 = 10 of the rest you don't have a ball in 3rd place; in 4C1 = 4 of the rest you don't have a ball in 4th place; in a single remaining case all four of your balls are closer than your opponent's closest.
Therefore your expected score is 20*1 + 10*2 + 4*3 + 1*4) / 35 = 56/35 = 1.6.
For the extension, the same process can be applied. You end up with
( (14C7)*1 + (13C6)*2 + (12C5)*3 + (11C4)*4 + (10C3)*5 + (9C2)*6 + (8C1)*7 + (7C0)*8 ) / (15C7) = 11440 / 6435 = 1.7777...
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Posted by tomarken
on 2021-06-02 11:35:58 |
Solution
