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CHAPTER 10
PROBABILITY
________________________________________________________________
Introduction:When a Fair coin is tossed, the face shown up is head or tail. Before tossing the coin, it is not
possible to predict which face shows up. Such experiments where the outcomes cannot be
predicted with certainty are called Random experiments. Random experiment and
phenomena are the heart of probability theory which owes its origin to the study of games of
chance or gambling.
B.PASCAL(1623-1662)and PIERRE de FERMAT(1601-1665), both French
mathematicians were the founding fathers of mathematical probability theory which is used
in many fields as Genetic engineering , Insurance and Medical sciences.
There are three possible states of expectation namely
(I)certainty
(II)impossibility
(III)uncertainty
The probability theory describes certainty by 1, impossibility by 0 and the various grades
of uncertainties by co-efficient ranging between 0 and 1
Before proceeding further let us defines terms associated with probability.
Experiment: An operation which can produce some well-defined out comes is known as
experiment.
Random Experiment: A Random experiment is one whose outcome is unpredictable i.e. an
experiment in which all possible outcomes are known and the exact outcome cannot be
predicted in advance, is called a random experiment.
Example: (I) Tossing a fair coin
(II)Throwing a true die
(III)Drawing a card from a well-shuffled pack of 52 playing cards.
In each of these experiments the possible outcomes are many and any one of these could be
the result of the experiment.
TRIAL AND EVENT
The single performance involved in a probability experiment is called a Trial and the
outcome is an Event.
For example: Tossing a coin repeatedly twice is a trial and turning up of HH,TT,HT and TH an
event.
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Outcome
The result of a Trial is called an outcome.
Experimental or empirical definition of probability of an event E.
P(E)=
‫܌܍ܖ܍ܘܘ܉ܐ ܜܖ܍ܞ܍ ܍ܐܜ ܐ܋ܑܐܟ ܖܑ ܛܔ܉ܑܚ܂ ܎ܗ ܚ܍܊ܕܝۼ‬
‫ܛܔ܉ܑܚܜ ܎ܗ ܚ܍܊ܕܝܖ ܔ܉ܜܗ܂‬
Classical Definition of probability of an event E
P(E)=
‫ ܗܜ ܍ܔ܊܉ܚܝܗܞ܉܎ ܛ܍ܕܗ܋ܜܝܗ ܎ܗ ܚ܍܊ܕܝۼ‬۳
‫ܜܖ܍ܕܑܚ܍ܘܠ܍ ܍ܐܜ ܎ܗ ܛ܍ܕܗ܋ܜܝܗ ܍ܔ܊ܑܛܛܗܘ ܔܔ܉ ܎ܗ ܚ܍܊ܕܝۼ‬
Let us find the probability for some of the events associated with experiments.
Example 1
Find the probability of getting a head when a fair coin is tossed once.
Solution:In the case of the experiment of tossing a fair coin once, the number of possible
outcomes is two-Head(H) and Tail (H)
Let E be the event ‘getting a head’. The number of outcomes favourable to ‘E’is 1
Therefore
P (E) =P (head) =
P (E)
=
‫ ܗܜ ܍ܔ܊܉ܚܝܗܞ܉܎ ܛ܍ܕܗ܋ܜܝܗ ܎ܗ ܚ܍܊ܕܝۼ‬۳
‫ܛ܍ܕܗ܋ܜܝܗ ܍ܔ܊ܑܛܛܗܘ ܔܔ܉ ܎ܗ ܚ܍܊ܕܝۼ‬
૚
૛
Example 2
A bag contains a red ball , a blue ball and a yellow ball, all the balls being of the same size.
Malavika takes out a ball from the bag- without looking into it. What is the probability that
takes out the (I) yellow ball?
(II) red ball? (III) blue ball?
Solution: Malavika takes out a ball from the bag without looking into it .So,it is equally likely that
she takes out any one of them.
Let’Y’ be the event ‘the ball taken out is yellow, ‘B’ be the event ‘the ball taken out is blue’ and R
be the event ‘the ball taken out is red’
Now, the number of possible outcomes=3
(I)The number of outcomes favourable to the event Y=1
૚
So P(Y) = ૜
૚
Similarly (II) P(R) = ૜
And
(III) P(B) =
૚
૜
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Note: An event having only one outcome of the experiment is called an elementary event.
In the above example P(Y)+P(R)+P(B)=1
Observe that the sum of the probabilities of all the elementary events of an
experiment is1.This is true in general also.
Example 3
Suppose we throw a true die once(I)What is the probability of getting a number
greater than4?(II)What is the probability of getting a number less than or equal to 4?
Solution:(I) Here let ‘E’ be the event getting a number greater than ‘4’.
∴ The number of possible outcomes is 6 ie 1,2,3,4,5 and 6, and outcomes favorable to E are 5
and 6. Therefore, the number of outcomes favourable to E is 2.
2
ଵ
So, P(E)=P(number greater than 4)= 6 =
ଷ
(II) Let ‘F’ be the event ‘getting a number less than or equal to 4
Number of possible outcomes =6
Outcomes favourable to the event F are 1,2,3,4.
So, the number of outcomes favourable to F is 4.
4
ଶ
∴ P(F) = 6 =
ଷ
Note:(1) The probability of an event which is impossible to occur is ‘O’. Such an event
is called ‘impossible event’
(2)The possibility of an event which is sure (or certain)to occur is 1.Such an event is
called ‘certain event’
(3)The probability of an event E is a number P(E) such that O ≤ P(E) ≤ 1
(4)For any event E, P(E)+P(Ē)= 1, Where Ē stands for ‘not E’. E and Ē are called
complementary events.
∴ P(Ē)= 1– P(E)
Example 4
What is the probability of getting a number 8 in a single throw of a true die?
Solution: Number of possible outcomes =6 .
Since no face of the die is marked 8, so there is no outcome favourable to 8, the number of
such outcomes is zero.
଴
So, P(getting 8)=଺ =0
∴ It is an impossible event . Probability is 0.
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Example 5
What is the probability of getting a number less than 7 in a single throw of a true die?
Solution: Since every face of a die is marked with a number less than7,it is sure that we will
always get a number less than 7 when it is thrown once. So, the number of favourable
outcomes is the same as the number of all possible outcomes, which is 6.
଺
∴ P(E)=P(getting a number less than 7) = ଺ =1
It is a sure or certain event and hence probability is 1.
Example 6
Two players Nadal and Federer,play a tennis match. It is known that the probability of
Nadal winning the match is 0.62. What is the probability of Federer winning the
match?
Solution: Let N and F denote the event that Nadal wins the match and Federer wins the
match, repectively.
The probability of Nadal’s winning =P(N)=0.62
∴ Probability of Federer’s winning =P(F)=1–P(N)
Here the events N and F are complementary events.
∴P(F) =1–P(N)
=1–0.62
=0.38
Example 7
A box contains 3 blue, 2 white and 4 red marbles. If a marble is drawn at random from
the box, what is the probability that it will be (i)white (ii)blue (iii)red ?
Solution: Saying that a marble is drawn at random is a short way of saying that all the
marble are equally likely to be drawn. Therefore, the number of possible
outcomes=3+2+4=9.
Let W denote the event ‘ the marble is white’, B denote the event ‘the marble is blue’ and
R denote the event ‘ the marble is red’.
(i)The number of outcomes favourable to the event W=2
∴P(W)=
૛
ૢ
૜
Similarly (ii) P(B)= ૢ
૝
(iii) P(R)= ૢ
=૜૚
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Example 8
Two true dice, one blue and one grey, are thrown at the sametime,
sametime write down all the
probability that sum of the two numbers appearing on the top of the dice is
(i)8 (ii) 13 (iii)) less than or equal to 12
Grey
Blue
So, the number of possible outcomes =6x6=36.
(i)The
)The outcomes favourable to the event ‘the sum of the two numbers is 8’denoted by E, are
(2,6),(3,5),(4,4)(5,3)(6,2)
i.e. the number of outcomes favourable to E=5.
Hence, P(E)=
૞
૜૟
(ii) The sum of two number is 13 is an impossible event.
So,
∴ P(F) =
)
(
=0
(iii) From the above table it is clear that, all the outcomes are favourable to the event G, ‘Sum
of two numbers ≤12’
So,
G is a certain or sure event
(
P(G)= ( = 1
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Exercise 10.1
I . Complete the following statements
1. Probability of an event E+ probability of the event ‘not E’= __________________________
2. The sum of the probability of all the elementary events of an experiment is
_____________________
3. The probability of an event that cannot happen is _________________________ such an event is
called _______________________________
4. The probability of an event that is certain to happen is __________________________ and an
event is called ________________________
5. The probability of an event is greater than or equal to __________________ and less than or
equal to ________________________________
6. If P(E) =0.05, then the probability of the event ‘not E’[P(Ē)]=________________________________
II. 1. A coin is tossed 500 times and we get head: 285 times, tail: 215 times. When a fair coin
is tossed at random, what is the probability of getting (i) a head (ii) a tail?
2. A true die is thrown 300 times and the outcomes are noted as given below.
Outcome
Frequency
1
60
2
72
3
54
4
42
5
39
6
33
When a die is thrown at random, what is the probability of getting a
(i) 1
(ii)5
(iii)6
(iv)3
(v)2
(vi)4
3. A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What
is the probability that the ball drawn is (I) red (II) not red
4. It is given that in a group of 3 students, the probability of 2 students not having the same
birthday is 0.992. What is the probability that the 2 students have the same birthday?
5. A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken
out of the box at random. What is the probability that the marble taken out will be
(I)red?
(II)white?
(III)not green?
6. A true die is thrown once. Find the probability of getting
(I) a prime number (II)a number lying between 2 and 6
(III)an odd number.
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7. A lot of 20 bulbs contains 4 defective ones. One bulb is drawn at random from the lot.
What is the probability that this bulb is defective?
8. A box contains 90 discs which are numbered from 1 to 90.If one disc is drawn at random
from the box, find the probability that it bears
(I)a two digit number(II)a perfect square number(III)a number divisible by 5
9. A jar contains 24 marbles, some are green and others are blue. If a marble is drawn at
ଶ
random from the jar, the probability that it is green is . Find the number of the blue marbles
ଷ
in the jar.
10. Tickets numbered 1to 20 are mixed together and then a ticket is drawn at random. What
is the probability that the ticket has a number which is multiple of 3 or 7?
11. A coin is tossed repeatedly twice .Find the probability of each event given below.
(i) A= The faces shown up are identical
(ii) B=The faces shown up are different
(iii)C=Head appears only once
(iv)D=Tail appears at least once
(v) E=Tail appears at most once
(vi)F=getting tail both times.
12. A True die is thrown once, what is the probability that it shows a number greater than3?
13. The probability that it will rain today is 0.84.What is the probability that it will not rain
today?
14.Find the probability that a number selected at random from the numbers 1 to 25 is not a
prime when each of the given numbers is equally likely to be selected.
15.A bag contains 4 white and 5 blue balls .They are mixed thoroughly and one ball is drawn
at random. What is the probability of getting (i)a white ball? (ii)a blue ball?
16.A bag contains 5 white, 6 red and 4 green balls. One ball is drawn at random. What is the
probability that the ball drawn is (i) green? (ii) white? (iii)non-red?
17. In a lottery, there are 10 prizes and 20 blanks. A ticket is chosen at random. What is the
probability of getting a prize?
18.It is known that a box of 100 electric bulbs contains 8 defective bulbs. One bulb is taken
out at random from the box. What is the probability that the bulb drawn is (i) defective
(ii)non-defective?
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19. A true die is thrown at random. Find the probability of getting (i)2 (ii) a number
less than 3(iii)a composite number
20.A box contains 19 balls bearing numbers 1,2,3, . . . . . . . . . . . .19 respectively.A ball is
drawn at random from the box. Find the probability that the number on the ball is
(i)prime number(ii)an even number (iii) a number divisible by 3.
21.Three fair coins are tossed simultaneously 150 times and it is found that 3 tails appeared
24 times, 2 tails appeared 45 times,1 tail appeared 72 times and no tail appeared 9 times.
If three coins are tossed simultaneously at random, find the probability of getting
(i)3 tails,(ii)2 tails, (iii)1 tail, (iv) 0 tail.
22.When 2 fair coins are tossed simultaneously, what are all possible outcomes? In a single
throw of 2 coins, what is the probability of getting (i) both heads?(ii)at least 1 head?
23. There are 35 students in a class, of whom 20 are boys and 15 are girls. From these
students one is chosen is at random. What is the probability that the chosen student is a
(i) boy? (ii) girl?
24.A survey of 250 girls of a school was conducted and it was found that 105 girls like tea
while 145 dislike it. Out of these girls, one girl is selected at random. What is the probability
that the selected girl (i)likes tea?(ii)does not like tea?
25.In a cricket match, a batsman hits the boundary 5 times out of 40 balls played by him.
Find the probability that the boundary is not hit by the ball.
26.The percentages of marks obtained by a student in six unit tests are given below.
Unit test
Percentage of marks obtained
I
53
II
72
III
28
IV
46
V
67
VI
59
A unit test is selected at random. What is the probability that the student gets more than
60% marks in the test?
27. The following table shows the blood groups of 40 students of a class
Blood group
Number of students
A
11
B
9
O
14
AB
6
One student of the class is chosen at random. What is the probability that the chosen
student has blood group (i) O? (ii)AB?
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Answers
Exercise 10.1
I. 1)1
2)0
5) 0,1
3) 0, impossible event
4)1, certain or sure event
6)0.95
II. 1) (i)0.57 (ii)0.43
2) (i)0.2 (ii)0.13
3) (i)
૜
ૡ
૚
6) (i) ૛
૚
(iii) ૞
૜
(iv)
૝
ૢ
૛
18)(i) ૛૞
(ii)
ૢ
૚ૢ
૚
22) ૝
૚
(iv)0.18
4)0.008
5) (i)૚ૠ
(iii) ૛
9)8
10)
૜
(vi)
૝
૞
(ii) ) ૢ
૛૜
(ii) ૛૞
(iii)
૞
૚
(ii) ૛
(v)
૝
15)(i)
૞
(ii)ૡ
(iii)0.11
૟
૚ૢ
૜
23) ૝
૚
૚
12)
૝
૝
16)(i) ) ૚૞
૚
૚
૛
૚
૛
૚
(ii) ) ૜
૚
(ii)
(vi)0.14
૚૜
(iii) ૚ૠ
8)(i)૚૙
11)(i)
૞
ૡ
(ii) ૚ૠ
ૢ
7) ૞
૛
(v)0.24
૚
૚
(ii) ૚૙
(iii) )
૛
13)0.16
૛
૚૟
14) ) ૛૞
૜
૚
(iii) ) ૞
૚
૚
17) ૜
ૡ
19)(i) ૟
(ii) ૜
(iii) ૜
20)(i) ૚ૢ
21)(i)0.16
(ii)0.30
(iii)0.48
(iv)0.06
૝
24)(i) ૠ
૜
(ii) ૠ
ૠ
25) ૡ
27)(i)0.35 (ii)0.15
************************************
૚
26) ૜