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

Probabilistic and Statistical Techniques Example = =

Probabilistic and Statistical Techniques Example

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 = 0.6545.

Probabilistic and Statistical Techniques Example

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

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

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

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

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

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

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?

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)

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?

P (Died of Cancer/ Current Cigar Smoker) = .00103/ .057= 0.018 Probabilistic and Statistical Techniques From a very large sample P (Died of Cancer) = 0.00739 P (Died of Cancer/ Current Cigar Smoker) = .00103/ .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

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.

Probabilistic and Statistical Techniques

    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’) =    

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 ?

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 ?

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