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### Probability C Hits the Target First

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Date: 09/08/2001 at 12:18:12
From: rob
Subject: Statistics/probability

A, B and C shoot at a target. A shoots first, followed by B, then by
C. They keep shooting in this sequence: A,B,C,A,B,C... until the
target is hit. A hits the target 5/6 of the time, B hits the target
3/4 of the time, and C hits the target 2/3 of the time. Find the
probability that C is the first one to hit the target.

Once I understand how to do this problem I'm sure others will make
sense to me; I just don't know where to start and what to do.
Thank you so much!

Rob
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Date: 09/08/2001 at 13:35:50
From: Doctor Mitteldorf
Subject: Re: Statistics/probability

Dear Rob,

This is indeed a tricky problem - but your experience with algebra
will serve you. I'm going to tell you what to do, then I'll let you
actually do the work. Of course, if you need more hints, or if you get
an answer you're not sure of, feel free to write back and I'll try to
respond as quickly as I can...

Here's what to do:

Analyze the first cycle exhaustively, with all the probabilities. The
probability that A misses is 1/6. Multiply that by the probability
that B misses, and multiply the product by the probability that C HITS
the target. Now you have the probability that C will be the first
one to hit the target IN THE FIRST CYCLE.

The next thing you might think of is: Now calculate the second and
third and fourth cycles the same way. Surely we'll see a pattern, and
the probabilities are getting smaller rapidly. This process must
converge, and maybe we'll get an infinite series that I know how to
sum...

Well, that's right - and that way would work, but here's a shortcut.
Just above, we calculated the probability that C would win in the
first cycle. That's a number that you know. We'll call that number c.
Now let p be the probability that C wins in ANY cycle. We'll be really
clever and write an equation that connects p to c. Here's how:

p can be broken down into two parts. One is the probability that C
wins in the first cycle, which is c. So p = c + something. The second
part is the probability that he wins in cycle 2, 3, 4, or any other
cycle. Here's the trick: if C fails to win the first cycle, all three
contestants are exactly back where they started. So we can write an
expression for the second part of p: it is (1-c) times p. In other
words, the equation you will write says, either C wins in the first
round, with a probability c which we know, or else C loses in the
first round, with a probability (1-c), and then we're back where we
started from and, by definition, his probability of winning is p.

In this way, you'll construct an equation - not a formula for p, since
p will appear on both sides, but a linear equation for p which you can
solve easily enough with elementary algebra.

- Doctor Mitteldorf, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Probability

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