Tossing a Coin and Rolling a Die

Date: 11/14/2002 at 12:21:34
From: Sally
Subject: Probabilities

If you toss a coin and roll a die, what is the probability of 
obtaining:

a) heads and a five  b) heads or a five  c) tails or a two?

While the answer to a) seems apparent  P(head) = 1/2
                                       P(5)    = 1/6
                                       P(head and 5) = 1/2 x 1/6 = 1/2

I am stumped because it looks as if that would be the answer to all 
three problems. What difference does "and" and "or" make?

Thanks.

Date: 11/15/2002 at 10:59:36
From: Doctor Ian
Subject: Re: Probabilities

Hi Sally,

Let's look at all the possibilities:

  h1  h2  h3  h4  h5  h6  t1  t2  t3  t4  t5  t6

How many contains a heads _and_ a five?

  h1  h2  h3  h4  h5  h6  t1  t2  t3  t4  t5  t6
                  --

So p(heads and five) = 1/12.  How many contain a heads _or_ a five?

  h1  h2  h3  h4  h5  h6  t1  t2  t3  t4  t5  t6
  --  --  --  --  --  --                  --

So p(heads or five) = 7/12. Now, let's look at what's going on here.

Basically, 'or' means to add up possibilities.  So

  p(heads) = 6/12

and

  p(5) = 2/12

and when we add them up, we get 8/12, which is clearly wrong. So what
kind of addition is this?  

It's addition with the possibility of duplication! Let's use -- to
indicate heads, and ~~ to indicate fives:

  h1  h2  h3  h4  h5  h6  t1  t2  t3  t4  t5  t6
  --  --  --  --  --  --                  
                  ~~                      ~~

By simply adding, we're counting h5 twice. So the answer is

  p(h or 5) = p(h) + p(5) - p(h and 5)

Does that make sense? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/