Chapter 18: Probability 255
Theoretical Probability ........................................................................................................
Relative frequency is useful in estimating the probability of events, but you don’t always have to
do all that observing and recording. Sometimes you can make an estimate of the probability of
some event based on things you already know (or can reasonably assume.) If you toss a coin, you
know it has two sides—a head and a tail—and you assume that heads and tails are equally likely.
As a result, you expect that the probability of heads is^12 or 50 percent, and the probability of tails
is also 50 percent. If you have a bag containing 10 marbles, 3 red, 4 blue, 2 green, and 1 yellow,
and you are planning to draw out one marble without looking, you can reasonably expect that
the probability that it will be red is 103 or 30 percent, and the probability that it will be blue is 104
or 40 percent. In the same way, the chance of drawing a green marble will be 20 percent, and the
chance of a yellow will be 10 percent.
The probability of something happening is a number between zero and one that tells you how
likely the event is. That number can be written as a fraction, a decimal, or a percent. If the
probability is zero, the event is impossible. It can’t happen. If the probability is one, it’s absolutely
certain to happen. It’s a sure thing. If you draw one item out of that bag of ten marbles, the
probability that it’s a kitten is 0. The probability that it’s a marble is 100 percent, or 1.
Most probabilities are somewhere in between zero and one, because most events are neither
absolutely impossible nor absolutely certain. Probabilities are fractions, but you’ll often hear them
expressed as percentages (a 30 percent chance of rain) or as ratios (1 in 10 chance of such-and-
such happening). The probability of an event is a fraction that compares the number of ways the
event can happen to the total number of things that can happen.
PE # of ways E can happen
# of things that can happen
If you take a well-shuff led deck of cards and pick one card at random, the probability of choosing
an ace is
4
52
1
13
because there are 4 aces out of 52 cards in the deck. The probability that the
card will be a heart is
13
52
1
4
because 13 of the 52 cards are hearts. The probability that you will
choose the ace of hearts is
1
52
because there is only one ace of hearts in the deck.
Probability of Compound Events
Basic probability is pretty simple. The number of ways things can go right, over the number of
ways things can happen. But life is complicated. Things don’t always happen one at a time. In
fact, they rarely happen one at a time. Usually there are at least two things to think about, but
if you can handle them two at a time you can work your way through everything. It’s all about
whether you want one or the other, or you want both.
