The Language of Argument

2 4 7

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

This makes sense because there are forty-eight out of fifty-two ways of not

drawing an ace, and this reduces to twelve chances in thirteen.

Probabilities of Conjunctions

We often want to know not just how likely it is that one single event will

occur but, instead, how likely it is that two events will occur together in a

certain order. Here’s a simple rule for calculating probabilities in some such

cases:

Rule 2: Conjunction with Independence. Given two independent

events, the probability of their both occurring is the product of

their individual probabilities. Symbolically (where h 1 and h 2 are

independent):

Pr(h 1 & h 2 ) 5 Pr(h 1 ) 3 Pr(h 2 )

Here the word “independent” needs explanation. Suppose you randomly

draw a card from the deck, then put it back, shuffle, and draw again. In this

case, the outcome of the first draw provides no information about the out-

come of the second draw, so it is independent of it. What is the probability of

drawing two aces in a row using this system? Using Rule 2, we see that the

answer is 1/13 3 1/13, or 1 chance in 169.

The situation is different if we do not replace the card after the first draw.

Rule 2 does not apply to this case because the two events are no longer inde-

pendent. The chances of getting an ace on the first draw are still one in thir-

teen, but if an ace is drawn and not returned to the pack, then there is one

less ace in the deck, so the chances of drawing an ace on the next draw are re-

duced to three in fifty-one. Thus, the probability of drawing two consecutive

aces without returning the first draw to the deck is 4/52 3 3/51, or 1 in 221,

which is considerably lower than 1 in 169.

If we want to extend Rule 2 to cover cases in which the events are not

independent, then we will have to speak of the probability of one event oc-

curring, given that another has occurred. The probability that h 2 will occur

given that h 1 has occurred is called the conditional probability of h 2 on h 1 and

is usually symbolized thus: Pr(h 2 |h 1 ). This probability is calculated by con-

sidering only those cases where h 1 is true and then dividing the number of

cases within that group where h 2 is also true by the total number of cases in

that group. Symbolically:

Pr 1 h 2 kh 125

favorable outcomes where h 1

total outcomes where h 1

5

outcomes where h 1 and h 2

total outcomes where h 1

Using this notion of conditional probability, Rule 2 can be modified as

follows to deal with cases in which events need not be independent:

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