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### High School Mathematics - 213.4 Compliment of an Event

#### Compliment of E

If E is an event of sample space S and É is the event that E does not happen then
P(É)=1-P(E)
É is called the compliment of E
n(E) + n(É) = n(S)
P(E) + P(É) = 1
Consider the example
A room has 3 sockets for bulbs. From a collection of 8 bulbs out of which 4 are defective , 3 bulbs are selected at random and put in the sockets. Find the probability that the room is lit
Solution

As 3 bulbs are selected from 8 bulbs
n(S) = 8C3== 56 ways
Let A be the event that the room is lit
This gives rise to 3 cases
• Either 1 bulb is non-defective
• Either 2 bulbs are non-defective
• Either 3 bulbs are non-defective

This involves tedious calculations and lengthy work. Hence consider the event Á whih implies that the room is dark
Á = 4C3 == 4 ways
P(Á) = 4/56 = 1/14
so P(A) = 1- P(Á)= 1- 1/14 = 13/14

Solved Examples

1. From a pack of 52 cards, 3 cards are drawn at random. Find the probabilty that
a)at least 1 is a heart.
b)all are not hearts.

Solution

n(S) = 52C3
a)let A be the event that at least one is a heart
So Á will be the event that non is a heart
n(Á) = 39C3
P(Á)=39C3 /52C3
P(Á)=39x38x37/52x51x50 = 703/1700
So the required probability is
P(A) = 1- P(Á).
P(A) = 1-703/1700 = 997/1700

b) Let B be the event that all are not hearts.
So P(&Bacute;) will be the event that all are hearts.
n(&Bacute;) = 13C3
P(&Bacute;) = 13x12x11/52x51x50 = 11/850
So the required probability is
P(B) = 1 - 11/850 = 839/850

2. A committee of 4 is to be formed from 10 boys and 1 girl at random. Find the probability that the girl is included.

Solution

n(S) = 11C4
Let A be the event that the girl is included.
So Á will be the event that the girl is never included.
n(Á)= 10C4
P(Á)= 10x9x8x7 / 11x10x9x8 = 7/11 So the required probability is
P(A) = 1 - 7/11 = 4/11

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