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Year 10 Interactive Maths - Second Edition


Mutually Exclusive Events

A and B are said to be mutually exclusive events if they do not overlap. This means that A and B are mutually exclusive events such that if A occurs then B is excluded or if B occurs then A is excluded. That is, A and B cannot occur together.

The Venn diagram shows events A and B are mutually exclusive events.

Note:

Mutually exclusive events have no sample points in common.


Consider the experiment of throwing a die.

S = {1, 2, 3, 4, 5, 6}

Let A be the event that an odd number is obtained and B be the event that an even number is obtained. Then:

A = {1, 3, 5}, B = {2, 4, 6}, Therefore A intersection B = the null set

That is, A and B have no elements (sample points) in common. Hence A and B are mutually exclusive events, as shown in the following Venn diagram.

The Venn diagram shows events A and B are mutually exclusive events. Event A contains the elements 1, 3 and 5 and event B contains the elements 2, 4 and 6. The number of elements in the sample space is 6.

Now, Pr(A intersection B) = Pr(A and B) = Number of elements in A and B / Number of elements in S = 0 / 6 = 0


Addition Law of Probabilities

For the example under consideration:

Pr(A) = Number of elements in A / Number of elements in S = 3/6 = 1/2, Pr(B) = Number of elements in B / Number of elements in S = 3/6 = 1/2, Pr(A or B) = Number of elements in A or B / Number of elements in S = 6/6 = 1   ...(1)

Also, Pr(A) + Pr(B) = 1/2 + 1/2 = 1   ...(2)     From (1) and (2), we obtain:  Pr(A or B) = Pr(A) + Pr(B)


Note:

Pr(A or B) is also denoted by Pr(A U B).


In general:

1.  If A and B are mutually exclusive events, then Pr(A U B) = Pr(A) + Pr(B)

The Venn diagram shows events A and B are mutually exclusive events.

2.  If A and B overlap, then Pr(A U B) = Pr(A) + Pr(B) - Pr(A intersection B)

The Venn diagram shows the respective sample space for events A and B have some common elements.


Consider the experiment of throwing a die. As usual:


S = {1, 2, 3, 4, 5, 6}

Let the events be defined as follows:

A = the event that an even number is obtained; and
B = the event that a prime number is obtained.

So A = {2, 4, 6}, B = {2, 3, 5}, A intersection B = {2}, A U B = {2, 3, 4, 5, 6}

The Venn diagram shows events A and B have the sample space element 2 in common. The sample space elements 4 and 6 are exclusive to event A and the sample space elements 3 and 5 are exclusive to event B.


Pr(A U B) = Number of elements in A U B / Number of elements in S = 5/6   ...(1),          Pr(A) = 3/6, Pr(B) = 3/6, Pr(A intersection B) = 1/6

We notice that:

Pr(A) + Pr(B) - Pr(A intersection B) = 3/6 + 3/6 - 1/6 = 5/6

From (1) and (2), we obtain:

Pr(A U B) = Pr(A) + Pr(B) - Pr(A intersection B)

This is called the addition law of probabilities.


Example 6

A die is rolled. If A = {greater than 3} and B = {prime}, find Pr(A or B).

Solution:

S = {1, 2, 3, 4, 5, 6}, A = {4, 5, 6}, B = {2, 3, 5}. So, A intersection B = {5}.

Now, Pr(A U B) = Pr(A) + Pr(B) - Pr(A intersection B) = 3/6 + 3/6 - 1/6 = 5/6


Key Terms

mutually exclusive events, addition law of probabilities


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