Complementary Events

Complementary events are two outcomes of an event that are the only two possible outcomes. This is like flipping a coin and getting heads or tails.

 P(E) + P(E¯) = 1, where E and E¯ are complementary events.

The event E¯, representing ‘not E‘, is called the complement of the event E.

Example: 1

What is the probability of drawing a heart from a deck of cards?

Solution:

There are total 52 cards in a deck out of which 13 cards are of heart.

So the favourable outcomes are 13 and the total no. of events is 52.

\mathrm{P(E)=\dfrac{Number\ of\ outcomes\ favourable\ to\ F}{Number\ of\ all\ outcomes}}

=\dfrac{13}{52}=\dfrac{1}{4}

Example 2

If we toss two coins together, then what is the probability of getting at least one head ?

Solution:

If we toss two coins together then the total outcomes could be The favourable outcomes for at least one head will be

\mathrm{\{HH\}, \{HT\}, \{TH\} = 3}

P (for at least one head) = \dfrac{3}{4}

Example 3

Find the probability that a leap year selected randomly will have 53 Sundays?
Solution
No. of days in a leap year = 366 days = 52 weeks + 2 days
It implies a leap year will have 52 Sundays. In remaining 2 days, possible outcomes are:
Sun, Mon
on, Tue
Tue, Wed
Wed, Thu
Thu, Fri
Fri, Sat
Sat, Sun
Total out comes = 7  Favourable outcomes that Sunday will come in these two days = 2
Required probability = \dfrac{2}{7}

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