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## Kick-off

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

 Puzzle ID: #11311 Fun: (2.39) Difficulty: (2.45) Category: Probability Submitted By: jimbo Corrected By: Anupriya9

Jimbo is an ace goal kicker. On the average, he will kick a goal on 2 out of 3 occasions. Unco on the other hand, will normally only kick a goal on 1 out of 2 occasions.
Now Jimbo, being quick to spot an opportunity, challenges Unco to a kicking duel. They will each in turn take one kick for goal until a goal is scored. The winner will be the first to score a goal.
"I'll even give you the first kick", says Jimbo generously to Unco.

What is the probability that Jimbo wins the kicking duel?

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 griphook Mar 28, 2003 I don't particularly like probability teasers but this one is real cool. Congrats. jimbo Mar 29, 2003 Thanks Grip. It's so nice to see a positive comment. I should keep that in mind! Pheonix_down Apr 10, 2003 This was really easy, I liked it though, I was about to make a teaser like this one before I read this... now I'm out of ideas! brianz May 16, 2005 Great teaser. I liked it. Not so hard since I just submitted one kinda like this. I don't know how you summed up the series, but here's how you could have done it. You just have to sum up 1/2*2/3*(1/6^0+1/6^1+1/6^2...) or 1/3*(1/6^0+1/6^1+1/6^2...) (1/2 that Unco misses his first shot, 2/3 that Jimbo makes one of his shots when he gets one, and the 1/6^0 is if Jimbo gets to kick on his first try, 1/6^1 is if Jimbo gets to kick on his second try, 1/6^2 is if Jimbo gets to kick on his third try...). This can then be simplified to 1/3*6/5 by the rules for summing up geometric series which then reduces to 2/5, your answer. That was fun. tsimkin Dec 31, 2008 Another way to solve this is as follows: P(Unco) = 1/2 + (1/2)(1/3)(P(Unco)) This is because Unco either wins on his first shot (with p = 1/2), or misses, and then Jimbo misses, and then Unco is back where he started. This equation becomes: P(U) = 1/2 + 1/6*P(U) 5/6*P(U) = 1/2 P(U) = 3/5 so P(J) = 2/5 Excellent teaser. Jimbo Dec 31, 2008 Thanks tsimkin. An excellent method that you have suggested! Zag24 Feb 07, 2009 Good puzzle. I also used tsimkin's technique, but, of course, I made a stupid arithmetic mistake and got the wrong answer. Doh!