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### AND and OR in Probability

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Date: 03/02/98 at 16:52:49
From: Alex King
Subject: probability

If there are two hats, one of which contains six slips of paper
numbered 1 to 6, and the other of which contains 26 slips, each with a
different letter of the alphabet on it, what are your chances of
picking a "2" and an "M"? Will you please answer this question and
show me how you got the answer? Also, will you send me some other
problems like this and let me see if I can answer them on my own?

Thanks!

Sincerely,

Alex King
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Date: 03/03/98 at 12:57:13
From: Doctor Bill
Subject: Re: probability

Alex,

Usually, in probability, if you want the probability of two or more
things happening, you multiply their probabilities together. That is,
you want Event 1 to happen AND you want Event 2 to happen, also. In
your problem, you want to draw a "2" (event 1) AND you want to draw
an "M" (event 2). Find the probability of each and multiply them
together to get the probability of drawing a 2 AND drawing an M.

Since there are 6 slips of paper in the first hat, the probability of
drawing a "2" is 1/6. There are 26 slips of paper in the second hat,
so the probability of drawing an "M" is 1/26. Thus, the probability of
drawing a 2 AND drawing an M is (1/6) * (1/26), which is about 0.0064.

What if you want the probability of drawing a "B" and a "5"? Would
this problem be any different than the problem you asked?

Notice in your problem you want two things to happen and the word
"AND" was used. That's usually a hint that you want to multiply the
probabilities together. If you had needed only one OR the other to
happen -- that is, you needed only one of the two events (notice we
used the word "OR" here) -- then you would add the probabilities. For
example, suppose you wanted to draw a "2" OR a "3." Then the
probability would be the sum of the two probabilities, that is
(1/6) + (1/6), which is about 0.3333. If you wanted to draw an "M" OR a
"Q," the probability would be (1/26) + (1/26), which is about 0.0769.

If you wanted to draw a "2" OR an "M," you have to be careful. These
events are not mutually exclusive -- that is, we could draw both. When
we add the probabilities of drawing a "2" (1/6) with the probability of
drawing an "M" (1/26), we've counted the probability of drawing a "2"
AND an "M" twice, once in the (1/6) and again in the (1/26), so we have
to subtract it out once. Thus the probability would be the sum of the
two probabilities MINUS the product of the probabilities, that is
(1/6) + (1/26) - (1/6)*(1/26), which is about 0.1987. Notice that it is
quite a bit more likely that you would draw a "2" OR an "M" than it is
that you would draw a "2" AND an M.

In probability, the words "AND" and "OR" are kind of special, and they
usually mean multiply the probabilities (for AND) and add the
probabilities (for OR), respectively.

-Doctor Bill, The Math Forum
Check out our web site http://mathforum.org/dr.math/

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Associated Topics:
High School Probability

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