CHAPTER 3 Getting Statistical: A Short Review of Basic Statistics 31

Don’t use percentage numbers (0 to 100) in probability formulas.

Even though these rules of probabilities may seem simple when presented here,

applying them together in complex situations — as is done in statistics — can get

tricky in practice. Here are descriptions of the not rule, the and rule, and the or rule.»

» ^{The not rule: The probability of some event X not occurring is 1 minus the}

probability of X occurring, which can be expressed in an equation like this:

Prob not

Prob

X

X

1

So if the probability of rain tomorrow is 0.7, then the probability of no rain

tomorrow is 1 – 0.7, or 0.3.»

» ^{The and rule: For two independent events, X and Y, the probability of event X}

and event Y both occurring is equal to the product of the probability of each of

the two events occurring independently. Expressed as an equation, the and

rule looks like this:

Prob

and

Prob

Prob

X

Y

X

Y

As an example of the and rule, imagine that you flip a fair coin and then draw

a card from a deck. What’s the probability of getting heads on the coin flip and

also drawing the ace of spades? The probability of getting heads in a fair coin

flip is 1/2, and the probability of drawing the ace of spades from a deck of

cards is 1/52. Therefore, the probability of having both of these events occur is

1/2 multiplied by 1/52, which is 1/104, or approximately 0.0096 (which is — as

you can see — very unlikely).»

» ^{The or rule: For two independent events, X and Y, the probability of X or Y (or}

both) occurring is calculated by a more complicated formula, which can be

derived from the preceding two rules. Here is the formula:

Prob

or

Prob

Prob

X

Y

X

Y

1

1

1

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 1

1

0 167

1

0 167

.

.

, which works out to 1

0 833

0 833

.

.

, 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