1. Lecture 5 0f 6
TOPIC 10
Independent
Events

2. (a) understand independent events
Learning Outcomes:
At the end of the lesson, students should be able to:
(a) understand independent events
(b) find the probability of independent events.

3. Let A is the event of the ‘ result of flipping a fair coin’ and B is the event of the ‘ number obtained when a pair of dice is rolled’.
These two events are unrelated.
Obtaining a head on the coin will not influence the outcome of the dice.
Such events are said to be independent.

4. From the probability rule for conditional events,
Then, we have

5. Definition
If A and B are independent events, it means that the outcome of one event does not affect the outcome of the other, then
and
Thus,

6. if A and B are two independent events
Remark
If A and B are independent events, then
(A and B’), (A’ and B), and (A’ and B’) are independent events too i.e.

7. Example 1:
Suppose two events A and B are independent. Given P(A) = 0.4 and P(B) = 0.25.
Calculate:

8. A and B are independent
Solution:
A and B are independent

9. Example 2
A, B and C are three events such that A and
B are independent whereas A and C are
mutually exclusive .
Given P(A) = 0.4 , P(B) = 0.2 , P(C) = 0.3 and
P(B ∩ C ) = Find
P(A U B)
P( C | B )
P( C | A’)

10. A and B are independent
Solution: a)
P(A U B) = b)
P( C | B )

11. c)
P( C | A’) C A B

12. Example 3
A mathematics puzzle is given to three students Amin, Ali and Abu. From the past experience, known that the probabilities Amin, Ali and Abu will get the correct solutions are 0.65, 0.6 and 0.55 respectively. If three of them attempt to solve the puzzle without consulting each other, find the probability that:
the puzzle will be solved correctly by all of them.
only one of them will get the correct solution.

13. Solution:
Let A= the event that Amin answers correctly B
= the event that Ali answers correctly C
= the event that Abu answers correctly
P(A) = 0.65, P(B) = and P(C)=0.55

14. The event that the puzzle will be solved correctly
by all of them is the event
b) The events that only one of them will get the
correct solution will occur if one of the events
occurs and all these events are mutually exclusive.

15. Thus,

16. Example 4
There are 60 students in a certain college, 27 of them are taking Mathematics, 20 are taking Biology and 22 are taking neither Mathematics nor Biology.
Find the probability that a randomly selected student takes
(i) both Mathematics and Biology.
(ii) Mathematics only.
b) A student is selected at random. Determine whether the event ‘ taking Mathematics’ is statistically independent of the event ‘ taking Biology’ .

17. Solution: M B
20-x
27-x x 22
Let B= the event of taking Biology
M= the event of taking Mathematics
x = the number of students taking both
Biology and Mathematics

18. Total number of students ,n(S) =60
a) x + x x + 22 = 60
x = 9
(i)
(ii)

19. b)
Hence, the two events are independent.

20. The probability that Roy is late for college on any day is 0.15 and is
Example 5
The probability that Roy is late for
college on any day is 0.15 and is
independent of whether he was late on the
previous day. Find the probability that he
is late on Monday and Tuesday
b) arrives on time on one of these days

21. 0.15 0.85 0.15 0.15 0.85 0.85 LATE LATE ON TIME LATE ON TIME ON TIME
Solution:
0.15
LATE
0.85
LATE
0.15
ON TIME
0.15
LATE
0.85
ON TIME
0.85
ON TIME
Monday
Tuesday

22. P( late on Monday and Tuesday )
= (0.15)(0.15) =
P( arrives on time on one of these days )
= (0.15) (0.85) + (0.15)(0.85) = =

23. EXERCISE
1. D and E are two independent events such that
P(D) = 0.3 and P(E) = Find
a) P(DE) Answer : 0.12
b) P(DE) Answer : 0.58
2. Three events, A , B and C are such that A and B
are mutually exclusive and
P(A)=0.3, P(C) = 0.4 , P(AB) = 0.5 and
P(BC) = 0.54.
Calculate P(B) and P(BC) Answer : 0.2 , 0.06
Determine whether or not B and C are
independent events.
Answer : B and C are independent events

24. CONCLUSION
The Independent Probability