Tossing a Coin and Rolling a DieDate: 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/ |
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