2. Concept of Probability
You are rolling a die.
What are the possible outcomes (value of
the top face) if the die is rolled only once?
Outcomes are either 1 or 2 or 3 or 4 or 5
or 6.
Chance of occurring ‘1’ is one out of six.
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3. Concept of Probability
Probability that outcome of roll of a die is ‘1’ is
1/6.
P(Die roll result = 1) = 1/6
Simple EVENT
Outcome Probability
1 1/6
2 1/6
3 1/6
4 1/6
5 1/6
6 1/6
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4. Concept of Probability
What is the probability that outcome of
roll of a die is an even number.
Thisoutcome can occur if the roll result is
2 or 4 or 6. i.e. 3 ways.
Number of all possible result is 6.
P(Even number) = 3/6 = 0.5
Similarly probability of odd number is 3/6.
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5. Concept of Probability
A 1 3 5 2 4 6 B
P(A) = 3/6 = 0.5
P(B) = 3/6 = 0.5
P(A) + P(B) = 1 NOTE
•A & B are mutually exclusive.
P(A) = 1 – P(B)
•A & B are mutually exhaustive.
•Sum probability of all outcome is 1.
•Probability is always ≥ 0.
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6. What is Sample Spaces
Collection of all possible outcomes.
All six faces of a die:
All 52 cards in a deck:
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7. Events
Simple event
Outcome from a sample space with one
characteristic.
e.g.: A red card from a deck of 52 cards
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8. Visualizing Events
Red cards 26
Black cards 26
Total cards 52
P(A red card is drawn from a deck) = 26/52
= 0.5
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9. Events
Compound event
Involves at least two outcomes
simultaneously.
e.g.:An ace that is also red from a deck of
cards.
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10. Visualizing Events
Ace Others Total
Red 2 24 26
Black 2 24 26
Total 4 48 52
P(An Ace and Red) = 2/52
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11. Impossible Events
Impossible event
e.g.: One card drawn is ‘Q’ of Club & diamond
Also known as ‘Null Event’
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12. Joint Probability
P(An ‘Ace’ and ‘Red’ from a deck of cards)
Ace Others Total
Red 2 24 26
Black 2 24 26
Total 4 48 52
A = ‘Ace’
P(A & B) = ?
B = ‘Red’
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13. Joint Probability
The probability of a joint event, A and B:
P(A and B) = P(A ∩ B)
B
A A&B
No. of outcomes from A and B
=
Total No. of possible outcomes
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14. Compound Probability
Probability of a compound event, A or B:
P(A or B) = P(A U B)
B
A A&B
No. of outcomes from A only or B only or Both
=
Total No. of possible outcomes
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15. Compound Probability
P(A or B) = P(A U B)
= P(A) + P(B) – P(A ∩ B)
All Ace All Red
4 2
26
4 26 2 7
= + - = Addition Rule
52 52 52 13
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16. Compliment Set
In roll of a die
Set of all outcomes = {1,2,3,4,5,6}.
If A = {1} then Ac has {2,3,4,5,6}.
If B = Set of even outcomes = {2, 4, 6} then Bc
has {1,3, 5}.
In
a class if A = Set of students that have
passed, then Ac is = Set of students that
have not passed.
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17. Computing Joint Probability
A = Card drawn from deck is ‘Ace’
Ac = Card drawn from deck is not ‘Ace’
B = Card drawn from deck is ‘Red’
B c = Card drawn from deck is not ‘Red’
Event
Total
Event B Bc
A 2 2 4
Ac 24 24 48
Total 26 26 52
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18. Computing Joint Probability
A = Card drawn from deck is ‘Ace’
Ac = Card drawn from deck is not ‘Ace’
B = Card drawn from deck is ‘Red’
B c = Card drawn from deck is not ‘Red’
Event
c
Total
Event B B
A A∩B A∩Bc A
Ac Ac ∩ B Ac ∩ B c Ac
Total B Bc
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19. Computing Joint Probability
Event
Total
Event B Bc
A P(A ∩ B) P(A ∩ B c) P(A)
Ac P(Ac ∩ B) P(Ac ∩ B c) P (Ac)
Total P(B) P(B c) 1
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20. Conditional Probability
Finding probability of an event A, given that
event B has occurred.
This means that we need to find out probability
of occurrence of ‘Ace’ given that the card drawn
is ‘Red’.
Event
Total
Event B Bc
2
A 2 2 4 =
c 26
A 24 24 48
Total 26 26 52
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21. Conditional Probability
Theprobability of event A given that event
B has occurred
P(A ∩ B)
= This is known as
P(B) ‘Conditional Probability’
and denoted as:
P(A l B)
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22. Multiplication Rule
P(A ∩ B)
P(A ∩ B) = x P(B)
P(B)
= P(A l B) x P(B)
= P(B l A) x P(A)
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23. Statistical Independence
Events A and B are independent if the
probability of one event, A, is not affected
by another event, B
P(A l B) = P(A)
P(B l A) = P(B)
P(A and B) = P(A) x P(B)
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24. Bayes’s Theorem
P(B l A) x P(A)
P(A l B) =
P(B)
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25. Bayes’s Theorem
P(B l A) x P(A)
P(A l B) =
P(B)
P(B l A) x P(A)
=
P(B ∩ A) + P(B ∩ Ac)
P(B l A) x P(A)
=
P(B l A) x P(A) + P(B l Ac) x P(Ac)
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26. Bayes’s Theorem (General)
P(A l Bi) x P(Bi)
P(Bi l A) =
P(A l B1).P(B1) +…+ P(A l Bk).P(Bk) )
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27. Example
Fifty percent of borrowers repaid their loans. Out of those
who repaid, 40% had a college degree. Ten percent of
those who defaulted had a college degree. What is the
probability that a randomly selected borrower who has a
college degree will repay the loan?
Repaid loan Not repaid loan Total
College
degree 0.2 0.05 0.25
No college
degree 0.3 0.45 0.75
Total
0.5 0.5 1.0
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28. Example
Fifty percent of borrowers repaid their loans. Out of those
who repaid, 40% had a college degree. Ten percent of those
who defaulted had a college degree. What is the probability
that a randomly selected borrower who has a college degree
will repay the loan?
CD - College degree
NCD – No College degree
P(RL | CD) = ?
RL – Repaid Loan
NRL – Not Repaid Loan
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29. Example
Fifty percent of borrowers repaid their loans. Out of those
who repaid, 40% had a college degree. Ten percent of
those who defaulted had a college degree. What is the
probability that a randomly selected borrower who has a
college degree will repay the loan?
CD - College degree
P(RL | CD) = ?
NCD – No College degree
RL – Repaid Loan 0.4 0.5
NRL – Not Repaid Loan
P(CD | RL) P(RL)
P(CD | RL) P(RL) + P(CD | NRL) P(NRL)
0.4 0.5 0.1 0.5
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30. Class Exercise
1. P(A) = 0.25, P(B) = 0.4, P(A|B) = 0.15
Find out P(AUB).
2. Probability of two independent events A
and B are 0.3 and 0.6 respectively. What
is P(A∩B)?
3. P(A∩B) = 0.2 ; P(A∩C) = 0.3 ;
P(B|A) + P(C|A) = 1; What is P(A)?
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31. Class Exercise
4. A jar contains 6 red, 5 green, 8 blue and
3 yellow marbles.
a) What is the probability of choosing a red
marble?
5. You are tossing a coin three times.
a) What is the probability of getting two tails?
b) What is the probability of getting at least 2
heads?
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32. Class Exercise
6. A plant has 3 assembly lines that
produces memory chips. Line1 produces
50% of chips (defective 4%), Line2
produces 30% of chips (defective 5%),
Line3 produces the rest (defective 1%).
A chip is chosen at random from
produced lot.
a) What is the probability that it is defective?
b) Given that the chip is defective, what is the
probability that it is from Line2?
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33. Class Exercise
7. An urn initially contains 6 red and 4
green balls. A ball is chosen at random
and then replaced along with two
additional ball of same colour. This
process is repeated.
a) What is the probability that the 1st and 2nd
ball drawn are red and 3rd is green?
b) What is the probability of 2nd ball drawn is
red?
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34. Class Exercise
8. Two squares are chosen at random on a
chessboard. What is the probability that
they have a side in common?
9. An anti aircraft gun can fire four shots at
a time. If the probabilities of the first,
second, third and the last shot hitting
the enemy aircraft are 0.7, 0.6, 0.5 and
0.4, what is the probability that four
shots aimed at an enemy aircraft will
bring the aircraft down?
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