1. Probabilistic and Statistical Techniques
Lecture 10 Eng. Ismail Zakaria El Daour
2011

2. Probabilistic and Statistical Techniques
Example = =

3. Probabilistic and Statistical Techniques
Example

4. Example Solution Probabilistic and Statistical Techniques
A car repair is either on time or late and either satisfactory or unsatisfactory .If a repair is made on time, then there is a probability of 0.85 that it is satisfactory. There is a probability of 0.77 that a repair will be made on time . What is the probability that a repair is made on time and is satisfactory ?
Solution
Let the event O be an on time repair and let the event S be a satisfactory repair.
It is known that P(S | O) = 0.85 and P(O) = 0.77.
The question asks for P(O  S) which is
P(O  S) = P(S | O) × P(O) = 0.85 × 0.77 =

5. Probabilistic and Statistical Techniques
Example

6. Example What’s the Probability? Event C D Total A 4 2 6 B 1 3 5 10
Probabilistic and Statistical Techniques
Example
What’s the Probability?
P(A) =
P(D) =
P(C  B) =
P(A  D) =
P(B  D) =
Event C D
Total A 4 2 6 B 1 3 5 10

7. Solution The Probabilities Are: P(A) = 6/10 P(D) = 5/10
Probabilistic and Statistical Techniques
Solution
The Probabilities Are:
P(A) = 6/10
P(D) = 5/10
P(C  B) = 1/10
P(A  D) = 9/10
P(B  D) = 3/10
Event C D
Total A 4 2 6 B 1 3 5 10

8. Event Probability Using Two–Way Table
Probabilistic and Statistical Techniques
Event Probability Using Two–Way Table
Event
Event B B
Total 1 2 A
P(A  B )
P(A  B )
P(A ) 1 1 1 1 2 1 A
P(A  B )
P(A  B )
P(A ) 2 2 1 2 2 2
Total
P(B )
P(B ) 1 1 2

9. Conditional Probability Using Two–Way Table
Probabilistic and Statistical Techniques
Conditional Probability Using Two–Way Table
Experiment: Draw 1 Card. Note Kind, Color & Suit.
Color
Type
Red
Black
Total
Ace 2 4
Non-Ace 24 48 26 52

10. Example Using the table then the formula, what’s the probability?
Probabilistic and Statistical Techniques
Example
Using the table then the formula, what’s the probability?
P(A|D) =
P(C|B) =
Are C & B Independent?
Event C D
Total A 4 2 6 B 1 3 5 10

11. Solution Using the formula, the probabilities Are: P(A  D) 2 / 10 2
Probabilistic and Statistical Techniques
Solution
Using the formula, the probabilities Are:
P(A  D) 2 / 10 2
P(A | D) = = =
P(D) 5 / 10 5
P(C  B) 1 / 10 1
P(C | B) = = =
P(B) 4 / 10 4 5 1 ≠
P(C) =
= P(C | B)
Dependent 10 4

12. Example Probabilistic and Statistical Techniques
Suppose that we have two events, A and B with P(A)=0.5, P(B)=0.6 and P(A &B)=0.4
Find:
P(A/B)
P(B/A)
Are A and B independent events? Why?

13. Example Probabilistic and Statistical Techniques
Assume that we have two events, A and B that are disjoints, assume also that P(A)=0.3, P(B)=0.4
What is (A & B)
What is P(A & B)
What is P(A/B)

14. Probabilistic and Statistical Techniques
From a very large sample
Relative Frequencies
Died of
Cancer
Did Not Die
of Cancer
Totals
Never Smoke Cigars
.00570
.880
.886
Former Cigar Smoker
.00066
.057
Current Cigar Smoker
.00103
.056
.00739
.993
1.000
Is Died of Cancer independent of cigar smoking?

15. P (Died of Cancer/ Current Cigar Smoker) = .00103/ .057= 0.018
Probabilistic and Statistical Techniques
From a very large sample
P (Died of Cancer) =
P (Died of Cancer/ Current Cigar Smoker) = / .057= 0.018
Since P (Died of Cancer) does not equal P (Died of Cancer/ Current Cigar Smoker)
So the events are dependent
Lecture 11

16. Example Probabilistic and Statistical Techniques
In a survey of MBA students, the following data were obtained on ‘students’ first reason for application to the school in which they joined.

17. Probabilistic and Statistical Techniques

18.     What can you say about: - P(A B) + P(A B’) =
Probabilistic and Statistical Techniques
What can you say about:
- P(A B) + P(A B’) =
- P(A B) + P(B A’) =    

19. Probabilistic and Statistical Techniques
Example
A bag contains 200 balls that are red or blue and either dull or shiny. There are 55 shiny red balls, 91 shiny balls, and 70 red balls. If a ball is chosen at random.
1- Find P( either shing or red )?
2- Find P( dull and bule )?
3- What is the probability of the chosen ball being shiny conditional on it being red ?
4-What is the probability of the chosen ball being dull conditional on it being red ?

20. Probabilistic and Statistical Techniques
Example
A bag contains 150 balls that are red or blue and either dull or shiny. There are 36 shiny red balls, 54 blue balls. If a ball is chosen at random.
What is the probability of the chosen ball being shiny conditional on it being red ?
What is the probability of the chosen ball being dull conditional on it being red ?

21. Thanks for Your Attention