1. TARUNGEHLOTS
Probability
In our everyday life, we often come across events which involve some sort of idea of the likelihood or
chance of occurrence.
When people ask, ‘Is it likely that a particular event will occur?’ We are concerning about the chance that
such an event occurs. In mathematics, the chance of occurrence of an event can be measured by a number
called the probability.
For any event E, the probability of its occurrence, denoted by P ( E ) , is defined as:
N u m b er o f o u tco m es favo u rab le to th e eve n t
P(E )
T o tal n u m b er o f p o ssib le o u tco m es
If an event E is certain to happen, the probability of its happening is 1.
If an event E is certain not to happen, the probability of its happening is 0.
Thus the probability of any event E is always lies between 0 and 1.
i.e. 0 P(E ) 1
Simple examples
Example 1
If we throw a fair die, any one of the six numbers 1, 2, 3, 4, 5 and 6 may come up. There are altogether 6
possible outcomes and the occurrence of each of these outcomes is equally likely. Hence, we conclude
that each number has an equal chance to appear and the probability of throwing, say, the number ‘1’ is
1
equal to .
6
1
The probability for the event: ‘number 5 comes up’ is denoted by P (5) . In fact, P (5) .
6
Think!
In a single throw of a die, what is the probability of getting
(a) an odd number?
(b) an even number?
(c) a prime number?
(d) a number greater than 4?

2. (e) a number from 1 to 6?
(f) a number except 5?
Answer:
(a) From 1 to 6, there are 3 odd numbers: 1, 3 and 5.
3 1
P (an odd number)
6 2
(b) From 1 to 6, there are 3 even numbers: 2, 4 and 6.
3 1
P (an even number)
6 2
(c) From 1 to 6, there are 3 prime numbers: 2, 3 and 5.
3 1
P (a prime number)
6 2
(d) From 1 to 6, there are 2 numbers greater than 4: 5 and 6.
2 1
P (a number greater than 4)
6 3
(e) From 1 to 6, there are 6 numbers that are from 1 to 6: 1, 2, 3, 4, 5 and 6.
6
P (a number from 1 to 6) 1
6
(f) From 1 to 6, there are 5 numbers that are not equal to 5: 1, 2, 3, 4 and 6.
5
P (a number except 5)
6
Example 2
If a card is drawn from a pack of 52 playing cards, any one of 52 cards may come up. There are altogether
52 possible outcomes and the occurrence of each of these outcomes is equally likely.
Think!
If we draw a card from a pack of 52 playing cards, what is the probability of getting
(a) the ace of diamonds?
(b) a card of heart?
(c) a card of number 8?
Answer:
(a) In a pack of playing cards, there is only one card of the ace of diamonds.
1
P (ace of diamonds)
52
(b) There are altogether 13 cards of heart.

3. 13 1
P (heart)
52 4
(c) There are altogether 4 cards of number 8.
4 1
P (8)
52 13
Example 3
When a baby is born, the probability that it is a boy or a girl is theorectially equal,
1
i.e. P (boy) P (girl)
2
Think!
If a woman has three daughters, what is the probability that the next baby is a boy?
Answer:
Note that existing fact does not affect the probability, so the probability that the next baby is a boy is still
1
.
2
Probability concerning ‘or’ , ‘and’
P ( E and F ) P(E ) P(F )
P ( E or F ) P(E ) P(F ) P ( E and F )
Think!
When a card is drawn from a pack of 52 playing card, what is the probability of getting
(a) an ace and a heart?
(b) an ace or a heart?
Answer:
4 13 1
(a) P (ace and heart) P (ace) P (heart)
52 52 52
4 13 1 16 4
(b) P (ace or heart) P (ace) P (heart) P (ace and heart)
52 52 52 52 13