calculated by a more complicated formula, which can be derived from the preceding two rules.

Here is the formula:

As an example, suppose that you roll a pair of six-sided dice. What’s the probability of rolling a 4

on at least one of the two dice? For one die, there is a 1/6 chance of rolling a 4, which is a

probability of about 0.167. (The chance of getting any particular number on the roll of a six-sided

die is 1/6, or 0.167.) Using the formula, the probability of rolling a 4 on at least one of the two

dice is

, which works out to

, or 0.31, approximately.

The and and or rules apply only to independent events. For example, if there is a 0.7 chance

of rain tomorrow, you may make contingency plans. Let’s say that if it does not rain, there is a 0.9

chance you will have a picnic rather than stay in a read a book, but if it does rain, there is only a

0.1 chance you will have a picnic rather than stay in a read a book. Because the likelihood of

having a picnic is conditional on whether or not it rains, raining and having a picnic are not

independent events, and these probability rules cannot apply.

Comparing odds versus probability

You see the word odds used a lot in this book, especially in Chapter 13, which is about the fourfold

cross-tab (contingency) table, and Chapter 18, which is about logistic regression. The terms odds and

probability are linked, but they actually mean something different. Imagine you hear that a casino

customer places a bet because the odds of losing are 2-to-1. If you ask them why they are doing that,

they will tell you that such a bet wins — on average — one out of every three times, which is an

expression of probability. We will examine how this works using formulas.

The odds of an event equals the probability of the event occurring divided by the probability

of that event not occurring. We already know we can calculate the probability of the event not

occurring by subtracting the probability of the event occurring from 1 (as described in the

previous section). With that in mind, you can express odds in terms of probability in the following

formula:

With a little algebra (which you don’t need to worry about), you can solve this formula for probability

as a function of odds:

Returning to the casino example, if the customer says their odds of losing are 2-to-1, they mean 2/1,

which equals 2. If we plug the odds of 2 into the second equation, we get 2/(1+2), which is 2/3, which

can be rounded to 0.6667. The customer is correct — they will lose two out of every three times, and

win one out of every three times, on average.

Table 3-1 shows how probability and odds are related.