### Brain Teasers

# Kick-off

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

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?

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?

### Hint

You only get to take your next kick if the guy before you misses!### Answer

Unco kicks: probability of a miss = 0.5Jimbo kicks: probability of a win = 0.5 x 2/3 = 1/3

{Note Unco must have missed in order for Jimbo to get a kick at all}

Unco kicks: probability of a second miss = 0.5x1/3x 0.5 = 1/12

{Note both Unco and Jimbo must have missed for Unco to get a second kick}

Jimbo kicks: probability of goal = 1/12 x 2/3 = 1/18

Probability that Jimbo wins on first or second or third etc is:

P = 1/3 + 1/18 + 1/108 + ......

Which is a geometric series with limiting sum = 2/5

Answer: Probability that Jimbo wins = 2/5.

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