AND and OR in Probability

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

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/