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Duel (Posted on 2003-03-06) Difficulty: 3 of 5

One day your rival invites you out to an old fashion duel. The rules are as follows:

-You each take turns shooting each other.
-The duel stops when one person hits the other.
-Both of you are honourable enough to not take shots out of your turn.
-The probability of one person hitting the other is 1/2 and is independent of the probability of the other person hitting you.

Clearly, whoever shoots first has a distinct advantage. So your friends suggests flipping a coin for it. Little did you know that your rival uses a coin rigged in such a way that the probability of getting heads is only 1/3.

If you chose heads, what is the probability of you winning the duel?

  Submitted by np_rt    
Rating: 3.3333 (6 votes)
Solution: (Hide)

The first thing to do is to calculate the probability of winning if you shoot first.

Let P(n)=probability of hitting your rival on your nth turn if you shoot first.
P(1)=1/2
P(2)=(1/2)*(1/2)*(1/2)
The first 1/2 is you missing on your first try, the second is your rival missing, the third is you hitting him.
P(3)=(1/2)*(1/2)*(1/2)*(1/2)*(1/2)
...etc
In general, P(n)=(1/2)*((1/4)^(n-1))

The probability of you winning if you shoot first is the sum of all P(n)'s, which forms an infinite geometric series, assuming you guys can shoot forever, with the first term a=1/2 and the ratio r=1/4. Using the formula for the sum of this infinite geometric series (which converges), S=a/(1-r), you'd end up with a probability of 2/3. And consequently, the probability of winning for the person shooting second is 1/3.

Next there are two cases to consider, one where the coin lands heads and other tails.

P(winning)=P(heads)*P(going 1st)+P(tails)*P(going 2nd)
=(1/3)*(2/3)+(2/3)*(1/3)
=4/9

Now, if you were to choose tails, your probability of winning would be 5/9.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionPuzzle SolutionK Sengupta2008-05-14 15:53:26
AnswerK Sengupta2008-05-13 05:53:46
another methodCory Taylor2003-03-07 04:49:25
SolutionsolutionCharlie2003-03-07 04:08:18
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