Cambridge International Mathematics

We have previously used two-dimensional grids to represent sample spaces and hence find answers to certain

probability problems.

Consider again a simple example of tossing a coin and rolling a die simultaneously.

To determine the probability of getting a head and a ‘ 5 ’, we can

illustrate the sample space on the two-dimensional grid shown. We

can see that there are 12 possible outcomes but only one with the

property that we want, so the answer is 121.

However, notice that P(a head)=^12 , P(a ‘ 5 ’)=^16 and^12 £^16 = 121 :

This suggests that P(a headanda‘ 5 ’)=P(a head)£P(a ‘ 5 ’),

i.e., we multiply the separate probabilities.

INDEPENDENT EVENTS

It seems that ifAandBare two events for which the occurrence of each one does not affect the occurrence

of the other, then P(AandB)=P(A)£P(B).

The two events ‘getting a head’ and ‘rolling a 5 ’ are events with this property, as the occurrence or non-

occurrence of either one of them cannot affect the occurrence of the other. We say they areindependent.

If two eventsAandBareindependentthen P(AandB)=P(A)£P(B).

Example 10 Self Tutor

A coin is tossed and a die rolled simultaneously. Find the

probability that a tail and a ‘ 2 ’ result.

‘Getting a tail’ and ‘rolling a 2 ’ are independent events.

) P(a tailanda‘ 2 ’)=P(a tail) £P(a ‘ 2 ’)

=^12 £^16

= 121

COMPLEMENTARY EVENTS

Two events arecomplementaryif exactly one of themmustoccur.

The probabilities of complementary events sum to 1.

Thecomplementof eventEis denotedE^0. It is the event whenEfails to occur.

For any eventEwithcomplementaryeventE^0 ,

P(E)+P(E^0 )=1 or P(E^0 )=1¡P(E).

G COMPOUND EVENTS [10.4]

T

23456

H

1

coin

die

Probability (Chapter 25) 519

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