CK-12 Probability and Statistics - Advanced

http://www.ck12.org Chapter 3. An Introduction to Probability

We are being asked to calculate the probability that we will observe at least one head. You may find it difficult
to calculate since the heads will most likely occur very frequently during 10 consecutive tosses. However, if we
calculate the complement ofA, i.e., the probability thatno headswill be observed, our answer may become a little
easier. The complementA′is easy, it contains only one simple event:

A′={T T T T T T T T T T}

Since this is the only event that no heads appear and since all simple events are equally likely, then

P(A′) =

1

1024

Now, becauseP(A) = 1 −P(A′),then

P(A) = 1 −P(A′) = 1 −

1

1024

≈ 0. 999 = 99 .9%

That is a very high percentage chance of observing at least one head in ten tosses of a coin.

Lesson Summary

1. ThecomplementA′of an eventAconsists of all the simple events (outcomes) that arenotin the eventA.


2. TheComplementary Rulestates that the sum of the probabilities of an event and its complement must equal
   1, or for an eventA,P(A)+P(A′) = 1.

Review Questions

1. A fair coin is tossed three times. Two events are defined as follows:

A:{At least one head is observed}

B:{The number of heads observed is odd}

a. List the sample space for tossing a coin three times

b. List the outcomes ofA.

c. List the outcomes ofB.

d. List the outcomes of the eventsA∪B,A′,A∩B.

e. FindP(A),P(B),P(A∪B),P(A′),P(A∩B).

2. The Venn diagram below shows an experiment with five simple events. The two eventsAandBare shown.
   The probabilities of the simple events are:

P( 1 ) = 1 / 10 ,P( 2 ) = 2 / 10 ,P( 3 ) = 3 / 10 ,P( 4 ) = 1 / 10 ,P( 5 ) = 3 / 10.

FindP(A′),P(B′),P(A′∩B),P(A∩B),P(A∪B′),P(A∪B),P[(A∩B)′]andP[(A∪B)′].