Chapter 11.3, Problem 6LC

Solution

To identify: The difference between independent events and mutually exclusive events.

The two events, the numbers on the two dice rolled are equal and the sum is odd are mutually exclusive events.

Explanation:

The objective is to explain the difference between independent events and mutually exclusive events.

Independent Events are those in which occurring of one event does not affect occurring of second event

If C and D are independent events.

Then, for independent events

  $P\left(CandD\right)\text{= }P\left(C\right)\cdot P\left(D\right)$

  $P\left(CorD\right)\text{ =}PIC)+P\left(D\right)-P\left(C\right)\cdot P\left(D\right)$

For example,

A month is selected at random; a number from 1 to 30 is selected at random.

Both these events are independent as selection made from each does not affect the other

Selection of a month from months does not have any effect on selection of number from 1 to 30.

Mutually exclusive are those that cannot happen at the same time.

If C and D are mutually exclusive events.

Then

  $P\left(CandD\right)=0$

For mutually exclusive events:

  $P\left(CorD\right)=P\left(C\right)+P\left(D\right)$

For example:

Take two events, the numbers on the two dice rolled are equal and the sum is odd

First take the event C. is that numbers are equal

So. $C=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(4,4\right)\text{, }\left(5,5\right),\left(6,6\right)\right\}$

That means there are 6 favorable cases.

Therefore,

  $\begin{array}{c}P\left(C\right)=\frac{6}{36}\\ =\frac{1}{6}\end{array}$

Next take the second event D is that the sum is odd.

So, the sum of the two numbers can be any number from $3,5,7,9,11$

So,

  $\left\{\left(1,2\right),\left(2,1\right),\left(1,4\right),\left(4,1\right),\left(1,6\right),\left(6,1\right),\left(2,3\right),\left(3,2\right),\left(2,5\right),\left(5,2\right)\left(3,4\right),\left(4,3\right),\left(3,6\right),\left(6,3\right)\text{ ,}\left(4,5\right),\left(5,4\right),\left(5,6\right),\left(6,5\right)\right\}$ There are 18 favorable cases.

Therefore,

  $\begin{array}{c}P\left(D\right)=\frac{18}{36}\\ =\frac{1}{2}\end{array}$

In both of these cases none of the pair is common

So. $P\left(CandD\right)=0$

Therefore the two events, the numbers on the two dice rolled are equal and the sum is odd are mutually exclusive events