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February 21st, 2017, 06:14 AM  #1 
Newbie Joined: Jan 2017 From: London Posts: 10 Thanks: 0  Darts Probability
When a person started playing darts, his skill was such that on a standard board he had a 40% chance of hitting the number at which he aimed. When he was unsuccessful he hit either of the two numbers adjacent to his target with equal chances of 30%. He never scored doubles or trebles. Where did he aim to achieve the highest score for a typical three dart throw? After some practice he is now able to hit a double with one throw in five and a treble with one throw in ten. Should the person revise his strategy? I'm guessing he would aim the center, since the number of points you get will be highest there, even if he gets it around it? Also, I don't understand the second part of the problem... How is getting doubles and trebles going to affect his "strategy"? Last edited by Estermont; February 21st, 2017 at 06:18 AM. 
February 21st, 2017, 08:05 AM  #2 
Senior Member Joined: May 2016 From: USA Posts: 1,084 Thanks: 446 
I have some difficulty understanding the problem. Is it the case that there is zero probability of hitting the 25 or 50? What happens when a double or triple is missed: is the number aimed at hit? The problem does not seem well specified. The essence of the first part of the problem (assuming 0 probability of hitting the center or doubles and triples) involves calculating expected values. The expected value of aiming at the 20 is $0.4 * 20 + 0.3 * 1 + 0.3 * 5 = 8.0 + 0.3 + 1.5 = 9.8.$ The expected value of aiming at the 14 is $0.4 * 14 + 0.3 * 9 + 0.3 * 11 = 5.6 + 2.7 + 3.3 = 11.6.$ Would you aim at the 20 or the 14? 
February 24th, 2017, 06:39 AM  #3 
Newbie Joined: Jan 2017 From: London Posts: 10 Thanks: 0 
The fact that the problem is not specific enough is another problem... I don't understand it either. Can anyone help?

March 15th, 2017, 04:58 PM  #4 
Newbie Joined: Mar 2017 From: Usa Posts: 5 Thanks: 0 
Jeffm is on the right track. Since the numbers on a dart board are effectively randomly placed, there is no point in trying for a single long equation. You will need 20 calculations for part A and 20 new calculations for part B. It's not clear what happens when you miss a double or treble  you have to know how often you hit the double or treble of the adjacent number for part B. 

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