USING THE COMPLEMENT OF AN EVENT

The complement of an event is the set of all outcomes in the sample space that are not included in the event.

For example, in the event of rolling a 3 on a number cube, the complement is rolling any number other than 3, which means the complement is rolling a 1, 2, 4, 5, or 6.

We can apply probabilities to situations involving random selection, such as drawing a card out of a shuffled deck or pulling a marble out of a closed bag.

Example 1 :

There are 2 red jacks in a standard deck of 52 cards. What is the probability of not getting a red jack if you select one card at random ?

Solution : 

Step 1 :

P(event) + P(complement)  =  1

P(red jack) + P(not a red jack)  =  1

Step 2 :

The probability of getting a red jack is 2/52.

Substitute 2/52 for P(red jack).

2/52 + P(not a red jack)  =  1

Step 3 :

Subtract 2/52 from both sides.

[2/52 + P(not a red jack)] - 2/52   =  1 - 2/52

P(not a red jack)   =  52/52 - 2/52

P(not a red jack)   =  (52 - 2)/52

P(not a red jack)   =  50/52

Simplify.

P(not a red jack)   =  25/26

The probability that you will not draw a red jack is 25/26 . It is likely that you will not select a red jack.

Example 2 :

A jar contains 8 marbles marked with the numbers 1 through 8. If you pick a marble at random, what is the probability of not picking the marble marked with the number 5 ?

Solution : 

Step 1 :

P(event) + P(complement)  =  1

P(marble numbered 5) + P(not the marble numbered 5 ) = 1

Step 2 :

The probability of getting a marble numbered 5 is 1/8.

Substitute 1/8 for P(marble numbered 5).

1/8 + P(not the marble numbered 5)  =  1

Step 3 :

Subtract 1/8 from both sides.

[1/8 + P(not the marble numbered 5)] - 1/8   =  1 - 1/8

P(not the marble numbered 5)  = 8/8 - 1/8

P(not the marble numbered 5)   =  (8 - 1)/8

P(not the marble numbered 5)   =  7/8

The probability that you will not pick a marble numbered with 5 is 7/8 . It is likely that you will not pick a marble numbered with 5. 

Reflect

Why do the probability of an event and the probability of its complement add up to 1 ?

The complement is made up of all outcomes not in the event. When you put the outcomes of an event and its complement together, you get all possible outcomes of an event. The probability of getting all the possible outcomes equals 1.

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