1. • In statistics – as well as in real life, events are often categorized as either
dependent or independent.
• Dependent events influence the probability of other events – or their
probability of occurring is affected by other events.
• Independent events do not affect one another
4. i. Dependent Events
The occurrence of some events may affect the
probability of occurrence of others
A and B are two events from a sample space ‘S’. Such
that event A occur firstly and event ‘B’ occur secondly.
Then probability of event ‘B’ changes due occurring of
event ‘A’. These events are dependents events
5. Therefore probability of both events occur A and B
P ( A and B ) = P ( A ) x P ( B / A )
where
P ( B / A ) = Probability of event B given that event
‘A’ has occurred
6. Two events are independent if the result of the second
event is not affected by the result of the first event. If A and
B are independent events,
the probability of both events occurring is the product of
the probabilities of the individual events.
Independent Events
B
P
A
P
B
A
P
7. Independent Events
Two or more events are independent events if the
occurrence or nonoccurrence of one of the events
does not affect the occurrence or nonoccurrence
of the other events
B
P
A
P
B
A
P
i
.
'
'
. B
P
A
P
B
A
P
ii
B
P
A
P
B
A
P
iii
'
'
.
'
'
'
'
. B
P
A
P
B
A
P
iv
9. Example
There are 11 red and 11 blue
balls in a jar
Case -1 Without replacement
i. Two balls are drawn at random one by one
What is the probability 1st
ball is red and 2nd
ball is blue
Probability without replacement means once we draw an item, then
we do not replace it back to the sample space before drawing a
second item. In other words, an item cannot be drawn more than
once
10. Case -2 with replacement
ii. Two balls are drawn at random one by one
What is the probability 1st
ball is red and 2nd
ball is blue
11. A box contains six 10 Ω resistors and ten 30 Ω resistors. The resistors are all unmarked
and are of the same physical size. One resistor is picked at random from the box; find the
probability that:
i. It is a 10 Ω resistor. (ii) It is a 30 Ω resistor.
At the start, two resistors are selected from the box. Find the probability that:
iii. Both are 10 Ω resistors.
P(first selected is a 10 Ω resistor) = 6 / 16 = 3 / 8
P(second selected is also a 10 Ω resistor) = 5 /15 = 1 / 3
P(both are 10 Ω resistors) = 3 / 8 × 1 / 3 = 1 / 8
(ii) The first is a 10 Ω resistor and the second is a 30 Ω resistor.
(iii) Both are 30 Ω resistors.
(i) P(10 Ω) = 6 /16 = 3 / 8
(ii) P(30 Ω) = 10 /16 = 5 / 8
12. Example - 1
A vehicle contains two engines, a main engine and a backup. The
engine component fails only if both engines fail. The probability that
the main engine fails is 0.05 , and the probability that the backup
engine fails is 0.10.Assume that the main and backup engines
function independently. What is the probability that the engine
component fails?
13. Solution
The probability that the engine component fails is the
probability that both engines fail therefore
P(engine component fails ) = P( main engine fails and backup engine fails )
P(main engine fails and backup engine fails ) = P(main fails) x P( backup fails)
P(main engine fails and backup engine fails ) = P(main fails) x P( backup fails)
= 0.05 x 0.10 = 0.005 = 0.5 %
14. Example 2
Oil wells drilled in region A have probability 0.2 of
producing. Wells drilled in region B have probability 0.09 of
producing. One well is drilled in each region. Assume the
wells produce independently.
What is the probability both wells produce oil ? (0.018)
15. Example – 3
Mr. A speaks truth in 75%; cases and Mr. B in 80% of
the cases. In what percentage of cases are they likely
to contradict each other, narrating the same
incident? (Assume independent events ) choose the
correct answer
a. 5 % b. 15% c. 35% d. 45%
iv. At least one of them speaks the truth
ii. Probability of Same statement
iii. Probability either A or B Speak the truth
16. The event A1 that a certain part of an airplane works during a
flight is determined to be 0.99 .Sine the engineers are reluctant
to risk the probability 0.01 of failure they insert another pare in
parallel. This means the failure occurs if and only if both parts
fail. Let the event A2 stand for the success of the second part.
Assume that it has been determined that P ( A2 ) = 0.99 and that
A1 and A2 are independent events. Find the probability of
success.
17. a
b
The diagram shows a simplified circuit in which two
independent components A and B are connected in parallel.
The circuit functions if either or both of the components are
operational. It is known that if A is the event ‘component a is
operating’ and B is the event ‘component b is operating’ then
P(A) = 0.99, P(B) = 0.98 and P(A ∩ B) = 0.9702. Find the
probability that the circuit is functioning
18. Find the reliability of the system shown the following. First, identify the series
and parallel sub-systems. There are 4 sub-systems.
21. An electronic system consists of four components: A, B, C, and D. The
operation of the system can be represented by four switches, with A and B in
series, and C and D in parallel with the series combination of A and B, as
shown in Figure 1 below. Continuity between input and output means the
system is in operation. Assume that C and D are 95 % pairwise , A is 96 % and
B is 97%
22. A game consists of spinning an unbiased arrow on a square board and throwing an
unbiased die. The board contains the letters A, B, C and D. The board is so designed
that when the arrow stops spinning it can point at only one letter, and it is equally
likely to point at A , B, C or D. List all possible outcomes of the game, that is, of
spinning the arrow and throwing the die. Find the probability that in any one game
the outcome will be:
i. an A and a 6
ii. a B and an even number
iii. an A and an even number or a B and an
odd number
iv. a C and a number ≥ 4
A
B
C
D
23. Three missiles are fired at a target .If the probabilities of hitting the
target are 0.4, 0.5 and 0.6 respectively, and if the missiles are fired
independently what is the probability.
i that all the missiles hit the target (0.12)
ii. that exactly one hits the target (0.38)
iii. that exactly 2-hit the target (0.38)
iv. that atleast one of the three hits the target (0.88)
Example – 4
24. Assume events A, B and C are such that P ( A ) = 0.5 , P ( B ) = 0.6 , P ( C ) =
0.4, P ( A ∩ B ) = 0.3 , P ( B ∩ C ) = 0.2 , P ( C ∩ A ) = 0.1 and
P ( A ∩ B ∩ C ) = 0.05. Find
i.P ( A ∩B ∩ C’) by Venn Diagram ii. P ( A’ ∩ B ∩ C’) iii P (A U B U C)
P ( A ∩ B ∩ C )
A
B
C
0.25
P ( A ∩ B ∩ C’ )
0.15
P ( A ∩ B’ ∩ C )
0.05
0.05
P ( A’ ∩ B ∩ C )
0.15
P ( A’ ∩ B ∩ C’ )
0.15
P ( A ∩ B’ ∩ C’ )
0.15
P ( A’ ∩ B’ ∩ C)
25. A unit contains 7 Army and 5 Naval officers .Another contains 9 Army 4 Naval
officers. An officer is selected at random form the first unit and transferred in
the second unit. An officer is selected at random from the second unit. What
is the probability, that he is an Army Officer? and he
is a Naval officer?
Example – 5
26. A three-engine jet departed from Miami
International Airport en route to South America,
but one engine failed immediately after take
off. While the plane was turning back to the
runway, the other two -engines also failed, but
the pilot was able to make a safe landing. With
independent jet engines, the probability of all
three failing is only (0.0001)3
, OR
Independent Jet Engines Example
27. about one chance in a trillion .The FAA found that the
same mechanic who replaced the oil in all three
engines incorrectly positioned the oil plug sealing rings.
A goal in using three separate engines is to increase
safely with independent engines, but the use of a
single mechanic caused their operation become
dependent. Maintenance procedures now require that
the engines be serviced by different mechanics.
28. A coin is tossed and a single 6-sided die is rolled. Find the probability of landing on
the head side of the coin and rolling a 3 on the die.
29. Due to the extreme cost of interrupting production, a
manufacturer has two standby machines available in case a
particular machine breaks down. The machine in use has a
reliability of .94, and the backups have reliabilities of .90
and .80. In the event of a failure, a backup machine is brought
into service. If this machine also fails, the other backup is used.
Calculate the system reliability.
Solution
R1 = 0.94, R2 = .90, and R3 = .80 The system can be depicted in this way:
30. Example:
the chance of a flight being delayed is 0.2 (=20%),
what are the chances of no delays on a round trip
The chance of a flight not having a delay is 1 − 0.2 = 0.8,
so these are all the possible outcomes:
0.8 x 0.8 = 0.64 chance of no delays
0.2 x 0.8 = 0.16 chance of 1st
flight delayed
0.8 x 0.2 = 0.16 chance of return flight delayed
0.2 x 0.2 = 0.04 chance of both flights delayed
Solution
31. Example:
Two sets of cards with a letter on each card as follows are placed
into separate bags. randomly picked one card from each bag.
Find the probability that:
a) She picked the letters ‘J’ and ‘R’.
b) Both letters are ‘L’.
c) Both letters are vowels.
Solution:
a) Probability that she picked J and R =
b) Probability that both letters are L =
32. Probability Topics Tree
Probability Topics Tree
Random
Experiment
Sample
Space
Events
Probability
Outcomes Criteria Numeric
Random
Variable
Probability
Distribution
Expectation
Mutually Exclusive (Non Overlapping)
Non-Mutually Exclusive (Overlapping)
Independent
P( A B ) = P ( A ) P
( B )
Dependent
Conditional Probability
Counting Rules
42. • Where the vertical line is reads as given that and the
information above the vertical line describes the conditioning
event.
• If A and B are two events in a sample space S and if P ( B ) is
not equal to zero then the conditional probability of the event
A given that event B has occurred written as
Similarly
43. It should be noted that P (A / B ) satisfies all
the basic axioms of probability
44. Definition:
Def#1:
The conditional probability of the event is
the probability that the event will occur, provided
the information that an event B has already
occurred. This probability can be written as P(A|
B).
45. • A conditional probability is a probability whose
sample space has been limited to only those
outcomes that fulfill a certain condition.
• The conditional probability of event A given that
event B has happened is
P(A|B)=P(A ∩ B)/P(B).
• The order is very important do not think that
P(A|B)=P(B|A)! THEY ARE DIFFERENT.
46. Suppose a pair of dice is tossed once. If it is known that one die
shows a 3 What is the probability that other die shows a 6
1,1 2,1 3,1 4,1 5,1 6,1
1,2 2,2 3,2 4,2 5,2 6,2
1,3 2,3 3,3 4,3 5,3 6,3
1,4 2,4 3,4 4,4 5,4 6,4
1,5 2,5 3,5 4,5 5,5 6,5
1,6 2,6 3,6 4,6 5,6 6,6
S
A = { (1,3), (2,3), (3,1), (3,2), (3,3), (3,4) , (3,5), (3,6), (4,3), (5,3), (6,3)}
B = { (1,6), (2,6), (3,6), (4,6), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
47. What is conditional probability? A typical consumer survey might result in data
like that shown below
Type of Phone
Owned
Male
Consumers
Female
Consumers
Total for
each row
Basic Phone 651 660 1311
Smart Phone 1411 1021 2432
Total for each column 2062 1681 3743
i.P(consumer owning a Smartphone)
ii. If we define M: consumer is male S: consumers owns a Smartphone
P( S / M )
iii.Let F and B represent the events,
F: consumer is female B: consumer owns a basic phone P(B /F ) and
P(F/B)
48. A random sample of 200 students are classified according to class status and
smoking habits
1st
Year 2nd
Year 3rd
Year 4th
Year Total
Smokers 21 33 25 23 99
Non Smokers 47 26 29 9 101
68 59 44 29 200
If a student is selected at random from this group .Find the probability that
i.The student of 1st
year class given that the student is smoker (0.212)
ii.The student is non smoker given that the student of 4th
year (0.310)
49. In a certain college 25% of the students passed In Mathematics, 15 %
of the students passed Statistics and 10 % of the students passed both
Mathematics and Statistics. A Student is selected at random.
ii If he passed in mathematics, what is the probability that he passed in
statistics
i. If he passed statistics, what is the probability that he passed in
mathematics (0.666)
Editor's Notes
#44:In other words, conditional probability is the probability that an event has occurred, taking into account some additional information about the outcomes of an experiment
