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### Conditional Probability

```
Date: 12/31/97 at 14:21:01
From: chris Bonnette
Subject: Conditional probability

Thank you for providing such an interesting way of finding out math
information. It is much appreciated.

I tutor probability and statistics and people always seem to be very
confused by conditional probability:

P(AIB) = P(both events)/P(known event occurred) = P(AB)/P(B)

The confusion seems to lie in the use of the multiplication rule.
If the events are dependent, they seem to feel that they end up in an
infinite loop:

P(AB) = P(A)*P(BlA)...but P(BlA) = P(BA)/P(A).  It appears that
everything just cancels and you are left with P(AB) and are no closer
to finding the solution.  I'm looking for a clear and nonconfusing way
to initially teach this concept to them so that this confusion doesn't
even start.  Any suggestions?

Chris
```

```
Date: 12/31/97 at 17:21:18
From: Doctor Anthony
Subject: Re: Conditional probability

A picture of the sample space allows you to calculate conditional
probabilities without too much of the confusing notation.

Example:

A bag contains 5 similar coins except that one is double-headed.  A
coin is chosen at random and is tossed 4 times.  Each time it lands

Chosen coin is           Chosen coin is
prob = 1/5               prob = 4/5
----------------------------------------------------
(1/5) x 1               (4/5) x (1/2)^4      4 heads obtained.
----------------------------------------------------
xxxxxx                 xxxxxxxxxxx          other result.
-----------------------------------------------------

The sample space is restricted to the top line since we are told that
4 heads were obtained. We can find the probability that we are in the
lefthand box by putting its value over the total value of the top
line.

(1/5)
(1/5) + (4/5)(1/16)

1
=     -----------------
1   +   1/4

4
=       --------
4 + 1

=         4/5

So the probability that we have the double-headed coin, on the
evidence of throwing 4 heads in a row, is 4/5. Of course, even one
tail would have made it certain that we had a normal coin.

This method allows you to understand what is going on without the
somewhat opaque notation of conditional probability.

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Probability

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