The Language of Argument

2 5 1

S o m e ru l e s o f P r o b a b i l i t y

We need to calculate probabilities of combinations in a different way. For

example, if you flip a fair coin twice, what is the probability of its landing

heads and tails in any order? There are two ways for this to happen. The

coin could come up either heads then tails or tails then heads. These alterna-

tives are mutually exclusive, so the probability of this disjunction by Rule 3

is 1/4 1 1/4 5 1/2. This is confirmed by counting two possibilities (heads

then tails, tails then heads) out of four (heads then heads, heads then tails,

tails then heads, tails then tails). Another way to calculate this probability is

to realize that the first flip doesn’t matter. Whatever you get on the first flip

(heads or tails), you need the opposite on the second flip. You are certain to

get either heads or tails on the first flip, so this probability is 1. Then, regard-

less of what happens on the first flip, the probability of getting the opposite

on the second flip is 1/2. These results are independent, so the probability of

their conjunction by Rule 2 is the product 1 3 1/2 5 1/2.

We can also use our rules to calculate probabilities of combinations with-

out independence. Rule 2G tells us that the probability of drawing an ace,

not putting this card back in the deck, and then drawing a king is 4/52 3

4/51 5 16/2,652. But what is the probability of drawing an ace and a king in

any order? It is the probability of drawing either an ace or a king and then

drawing the other one, given that you drew the first one. That probability by

Rule 2G is 8/52 3 4/51 5 32/2,652. The difference between this result and

the previous one, where the order was specified, shows why we need to de-

termine whether we are dealing with permutations or combinations.

Use the rules of probability to calculate these probabilities:

1. What is the probability of rolling a five on one throw of a fair six-sided die?


2. What is the probability of not rolling a five on one throw of a fair six-sided die?


3. If you roll a five on your first throw of a fair six-sided die, what is
   probability of rolling another five on a second throw of that die?


4. If you roll two fair six-sided dice one time, what are the chances that both
   of the dice will come up a five?


5. If you roll two fair six-sided dice one time, what are the chances that
   one or the other (or both) of the dice will come up a five?


6. If you roll two fair six-sided dice one time, what are the chances
   that one and only one of the dice will come up a five?


7. If you roll two fair six-sided dice one time, what are the chances
   that at least one of the dice will come up a five?


8. If you roll two fair six-sided dice one time, what are the chances
   that at least one of the dice will not come up a five?


9. If you roll six fair six-sided dice one time, what are the chances
   that at least one of the dice will come up a five?


10. If you roll six fair six-sided dice one time, what are the chances that
    at least one of the dice will not come up a five?

exercise ii

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