1. Mathematics

2. Session
Probability -2

3. Conditional Probability
Session Objectives
Addition Theorem on Probability
Conditional Probability
Multiplication Theorem on Probability
Independent Events
Class Exercise

4. Addition Theorem For 2 Events
If A and B be are events in a sample space S. Then the probability
of occurrence of at least one of the events A and B is given by

5. Addition Theorem For 3 Events
If A, B and C are three events in the sample space S. Then,
If A, B and C are three mutually exclusive events, then

6. Addition Theorem For n Events

7. Event A or not A
The events ‘A’ and ‘not A’ are mutually exclusive events as each
outcome of the experiment is either favourable to ‘A’ or ‘not A’.
Therefore, the event ‘A or not A’ is a certain (or sure) event whose
probability is 1.

8. Example-1 Given two mutually exclusive events A and B such and
, find P(A or B).

9. Example-2
An integer is chosen at random from the first 200 positive integers.
Find the probability that the integer is divisible by 6 or 8.
Solution: Let S be sample space. Then,
S = {1, 2, 3, …200}, n(S) = 200
Let A : event that the number is divisible by 6.
A = {6, 12, }, n(A) = 33
Let B : event that number is divisible by 8.
B = {8, 16, }, n(B) = 25

10. A  B : event that the number is divisible either by 6 or 8.
Solution Cont.
(A Ç B) : event that the number is divisible by 6 and 8.
A Ç B = {24, 48, }, n(A Ç B) = 8
A  B : event that the number is divisible either by 6 or 8.

11. Example-3
Find the probability of drawing a black king and probability of drawing
a black card or a king from a well-shuffled pack of 52 cards.
Solution: Let A and B be two events such that
A : drawing a black card, B : drawing a king
Total number of cards n(S) = 52
Number of black cards n(A) = 26
Number of kings n(B) = 4

12. Solution (Cont.)

13. and P(drawing a black card or a king)
Solution (Cont.)
and P(drawing a black card or a king)

14. Solution: Let S be sample, then n(S) = 36
Example-4
Find the probability of getting an even number on the first die or a
total of 8 in a single through of two dice (CBSE 2004)
Solution: Let S be sample, then n(S) = 36
Let A and B be two events such that
A = getting an even number on first die, B = getting a total of 8

15. Solution Cont.
A ={(2, 1), (2, 2), …, (2, 6), (4, 1), (4, 2), …, (4, 6),
(6, 1), (6, 2), …, (6, 6)},
B = {(2, 6), (6, 2), (3, 5), (5, 3), (4, 4)}

16. Solution (Cont.)

17. Solution: Let S be sample, then n(S) = 36
Example-5
Two dice are tossed together. Find the probability that the sum of
the numbers obtained on the two dice is neither a multiple of 3 nor
a multiple of (CBSE 1996)
Solution: Let S be sample, then n(S) = 36
Let A and B be two events such that
A = the sum of the numbers obtained on the two dice is a multiple of 3.,
B = the sum of the numbers obtained on the two dice is a multiple of 4

18. Solution (Cont.)
Then, A = {(1, 2), (1, 5), (2, 1), (2, 4), (3, 3), (3, 6), (4, 2),
(4, 5), (5, 1), (5, 4), (6, 3), (6, 6)},
B = {(1, 3), (2, 2), (2, 6), (3, 1), (3, 5), (4, 4), (5, 3), (6, 2), (6, 6)}

19. Solution (Cont.)

20. Conditional Probability
In a random experiment, if A and B are two events, then the
probability of occurrence of event A when event B has already
occurred and , is called the conditional probability and it
is denoted by

21. Cont.

22. Multiplication Theorem on Probability
If A and B are two events associated with a random experiment, then

23. Solution: Given that P(A) = 0.5, P(B) = 0.6 and
Example-6
Solution: Given that P(A) = 0.5, P(B) = 0.6 and
By addition theorem

24. By multiplication theorem
Solution Cont.
By multiplication theorem

25. Example –7
A die is rolled twice and the sum of the numbers appearing on them
is observed to be 7. What is the conditional probability that the number
2 has appeared at least once (CBSE 1990, 2003)
Solution: Let A and B be two events such that
A = getting number 2 at least once
B = getting 7 as the sum of the numbers on two dice
A = {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
(1, 2), (3, 2), (4, 2), (5, 2), (6, 2)}

26. Solution (Cont.)
B = {(2, 5), (5, 2), (6, 1), (1, 6), (3, 4), (4, 3)}

27. Independent Events
Two events A and B are said to be independent if occurrence of one
does not affect the occurrence of other; then

28. Solution: (i) Let A : event of A passing the examination
Example –8
The probability of student A passing an examination is
and the probability of student B passing is . Find the probability of
only A passing the examination
(ii) only one of them passing the examination
Solution: (i) Let A : event of A passing the examination
P(A') = 1 – P(A)

29. Solution Cont. Let B : event of B passing the examination
P(B’) = 1 – P(B)
[As A and B' are independent events]

30. (ii) P(only one passing the examination)
Solution Cont.
(ii) P(only one passing the examination)
[As A and B' are independent events and A’ and B are also
independent events]

31. Example –9
Three bags contain 7 white 7 red, 5 white 8 red, and 8 white 6 red
balls respectively. One ball is drawn at random from each bag. Find
the probability that the three balls drawn are of the same colour.
Solution: W1 : White ball from bag I
W2 : White ball from bag II
W3 : White ball from bag III
R1 : Red ball from bag I
R2 : Red ball from bag II
R3 : Red ball from bag III

32. P(Three balls are of same colour)
Solution (Cont.)
P(Three balls are of same colour)
[As W1, W2 and W3 are independent events and R1, R2 and R3 are also
independent events]

33. Example –10
A speaks truth in 60% of the cases and B in 90% of the cases. In what
percentage of the cases they are likely to contradict each other in
stating the same fact? (CBSE 2003)
Solution: Let A : event of A speaking the truth.
B : event of B speaking the truth.

34. P(A and B contradicting each other)
Solution (Cont.)
P(A and B contradicting each other)
[As A and B' are independent of each other and A' and B are
also independent of each other.

35. Example –11 The probabilities of A, B and C solving a problem are
respectively. If the problem is attempted by all simultaneously,
find the probability of exactly one of them solving it. (CBSE 2004)

36. Solution (Cont.) Required probability
[As A, B’ and C’ are independent events; A’, B and C’ are independent
events; A’, B’ and C are also independent events]

37. Thank you