Multiplication Rule
Introduction
Sometimes we are interested in probability where there are two or more
independent events. Events A and B are independent when the happening of
one of them has no bearing on the happening of the other. To illustrate with
proof of facts, facts are independent for the purposes of proof when the truth
of Fact A has no bearing on the truth of Fact B. An illustration from gambling
is two successive throws of a dice. No matter what the result of the first
throw, it has no bearing on the second.
If we wish to calculate the probability that two or more independent events will
all occur we use the product or multiplication rule. This involves taking the
probability of each individual event and multiplying them together. We will deal
with this in stages, stating the main rule, then stating some subsidiary rules.
Main Rule
There are three aspects to the main version of the product rule -
derivation,
statement and illustration.
Derivation of Rule
Before we formally state this rule we will show how it is derived by using the
game of two-up. This involves throwing two coins into the air together, with a
spinning action, and betting on the outcome, that is, the sides of the coins that
face upwards. (In many other countries, the sides of a coin are labelled
“heads” and “tails” and we use this terminology here.) If we ignore
the very
remote chance that a coin can land on its edge there are two possibilities,
which are taken to be equally likely, heads with a probability of 50% and tails
with a probability of 50%.
Now consider throwing the two coins. There are four outcomes which are all
equally likely. We can set this out in the following table:
First Coin
Second Coin
Probability
Heads
Heads
25%
Tails
Tails
25%
Heads
Tails
25%
Tails
Heads
25%
100%
Figure 9.2 Derivation of the Multiplication Rule
How did we calculate the probability for each combination? We reasoned in
the following way, using the probability of two tails as an example:
(1)
When the first coin is thrown the chances are equal that it will come
down heads or tails, for example the probability of tails is 50%.
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