1. Probabilistic & Statistical Techniques Eng. Tamer Eshtawi First Semester 2007-2008 Eng. Tamer Eshtawi First Semester 2007-2008

2. Lecture 7 Chapter 4 (part 2) Probability Main Reference: Pearson Education, Inc Publishing as Pearson Addison-Wesley.

3. Section 4-4 Multiplication Rule: Basics

4. Key Concept If the outcome of the first event A somehow affects the probability of the second event B, it is important to adjust the probability of B to reflect the occurrence of event A. The rule for finding P(A and B) is called the multiplication rule.

5. Tree Diagrams A tree diagram is a picture of the possible outcomes of a procedure, shown as line segments emanating from one starting point. These diagrams are helpful if the number of possibilities is not too large. This figure summarizes the possible outcomes for a true/false followed by a multiple choice question. Note that there are 10 possible combinations.

6. Key Point – Conditional Probability The probability for the second event B should take into account the fact that the first event A has already occurred.

7. Notation for Conditional Probability represents the probability of event B occurring after it is assumed that event A has already occurred (read B /A as “B given A.”)

8. Definitions Independent Events Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. If A and B are not independent, they are said to be dependent.

9. Formal Multiplication Rule   Note that if A and B are independent events,

10. Applying the Multiplication Rule

11. Traffic Signal A box contains glass lenses used for traffic signal. 5 of the lenses are red, 4 are yellow and 3 are green. If 2 of the lenses are randomly selected. Fined the probability that: First is red and the second is green, Assume no replacement Example 1

12. Jury selection A pool of jurors consists of 10 men and 15 women, two names are selected randomly Find the probability that the first is a man and the second is a man a)With replacement b)Without replacement Solution The two events are independent a) b) Example 2

13. If two cards are drawn from a deck without replacement then the event that the first card drawn is a king the event that the second card is a king Find the probability of both events occurring is: Example 3

14. Small Samples from Large Populations If a sample size is no more than 5% of the size of the population, treat the selections as being independent (even if the selections are made without replacement, so they are technically dependent).

15. Recap In this section we have discussed:  Notation for P(A and B).  Notation for conditional probability.  Independent events.  Formal multiplication rules.  Tree diagrams.

16. Section 4-5 Multiplication Rule: Complements and Conditional Probability

17. Key Concept In this section we look at the probability of getting at least one of some specified event; and the concept of conditional probability which is the probability of an event given the additional information that some other event has already occurred.

18. Complements: The Probability of “At Least One”  The complement of getting at least one item of a particular type is that you get no items of that type.  “At least one” is equivalent to “one or more.”

19. Key Principle To find the probability of at least one of something, calculate the probability of none, then subtract that result from 1. That is, P(at least one) = 1 – P(none).

20. Definition A conditional probability of an event is a probability obtained with the additional information that some other event has already occurred.

21. Gender of Children Find the probability that a couple with 3 children having at least 1 girl. The gender of any child is not influenced by the next child gender Solution Example 1

22. MenWomenBoysGirlsTotal Survive d 3323182927706 Died136010435181517 Total169242264452223 Suppose a passenger on the Titanic is chosen at random, let M be the event that they are a man, W the event they are a woman, and S that they survived D that they died Calculate: Example 4

23. MenWomenBoysGirlsTotal Survive d 3323182927706 Died136010435181517 Total169242264452223 Solution

24. If two cards are drawn from a deck without replacement then the event that the first card drawn is a king the event that the second card is a king Find the following Example 5

25. And so the probability of at least one six occurring is: Example 6 If a die is rolled 4 times, find the probability that at least one of the rolls is a six. If E is the event that at least one roll is a six, then is the event that none of the four rolls is a six. Since all rolls are independent, the probability of four rolls all being not six is:

26. If two dice are rolled, then the event A of rolling a 7 and the event B of rolling an 11, so that the probability of rolling 7 or 11 is: If a card is drawn, then the event K that the card is a king and the event H that the card is a heart, so the probability of drawing a king or a heart is: Example 7

27. Recap In this section we have discussed:  Concept of “at least one.”  Conditional probability.

28. Flash points

29. If A and B are dependent events, then P(A and B) is A.P(A) P(B|A) B.P(A) P(B) C.P(A) P(A|B)

30. The following table contains data from a study of two airlines which fly to Small Town, USA. Number of on time flights Number of late flightsSum Airlines 133639 Airlines 243548 If one of the 87 flights is randomly selected. Find the probability that the flight selected arrived on time given that it was an Airlines 2 flight. A. 43/87 B. 11/76 C. 43/48 D. None of the above is correct.