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Lecture 09 Prof. Dr. M. Junaid Mughal

Mathematical Statistics. Lecture 09 Prof. Dr. M. Junaid Mughal. Last Class. Introduction to Probability Counting Rules Combinations Axioms of probability Examples. Today’s Agenda. Review of last lecture Probability (continued). Combinations.

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Lecture 09 Prof. Dr. M. Junaid Mughal

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  1. Mathematical Statistics Lecture 09 Prof. Dr. M. JunaidMughal

  2. Last Class • Introduction to Probability • Counting Rules • Combinations • Axioms of probability • Examples

  3. Today’s Agenda • Review of last lecture • Probability (continued)

  4. Combinations • Definition: When we have n different objects, and we want to have combinations containing r objects, then we will have nCr such combinations. (where, r is less than n).

  5. Probability • If there are n equally likely possibilities of which one must occur and s are regarded as favorable or success, then the probability of success is given by s/n • If an event can occur in h different possible ways, all of which are equally likely, the probability of the event is h/n: ClassicalApproach • If n repetitions of an experiment, n is very large, an event is observed to occur in h of these, the probability of the event is h/n: FrequencyApproach or EmpiricalProbability.

  6. Axioms of Probability • To each event Ai, we associate a real number P(Ai). Then P is called the probability function, and P(Ai) the probability of event Ai if the following axioms are satisfied • For every event A : P(Ai) ≥ 0 • For the certain event S : P(S) = 1 • For any number of mutually exclusive events A1, A2 …. An in the class C: P(A1U A2 U…..U An) = P(A1) + P(A2) + … + P(An) • Note: Two events A and B are mutually exclusive or disjoint if A  B = Φ

  7. Example • A die is loaded in such a way that an even number is twice as likely to occur as an odd number. If E is the event that a number less than 4 occurs on a single toss of the die, find P(E). • Sample Space

  8. Example • A die is loaded in such a way that an even number is twice as likely to occur as an odd number. If E is the event that a number less than 4 occurs on a single toss of the die, find P(E). • Sample Space

  9. Example • In the previous Example, let A be the event that an even number turns up and let B be the event, that, a number divisible by 3 occurs. Find P(A U B) and P(A B).

  10. Example • A statistics class for engineers consists of 25 industrial, 10 mechanical, 10 electrical, and 8 civil engineering students. If a person is randomly selected by the instructor to answer a question, find the probability that the student chosen is (a) an industrial engineering major, (b) a civil engineering or an electrical engineering major.

  11. Example • In a poker hand consisting of 5 cards, find the probability of holding 2 aces and 3 jacks.

  12. Some Theorems • For A or B two events, P(A U B) = P(A) + P(B) – P(A  B) • For mutually exclusive events P(A U B) = P(A) + P(B) • True for any number of events

  13. Theorems • If A1, A2, …., An are partitions of a sample space then P(A1 U A2 U …. U An) = P(A1) + P(A2) + … + P(An) = P(S) = 1

  14. Example • After being interviewed at two companies John assesses that his probability of getting an offer from company A is 0.8, and the probability that he gets an offer from company B is 0.6. If on the other hand, he believes that the probability that he will get offers from both companies is 0.5, what is the probability that he will get at least one offer from these two companies?

  15. Example • What is the probability of getting a total of 7 or 11 when a pair of fair dice are tossed?

  16. Example • If the probabilities are, respectively, 0.09, 0.15, 0.21, and 0.23 that a person purchasing a new automobile will choose the color green, white, red, or blue, what is the probability that a given buyer will purchase a new automobile that comes in one of those colors?

  17. Theorems (Cont.) • If A and A’ are complementary events then P(A) + P(A’) = 1

  18. Example • If the probabilities that an automobile mechanic will service 3, 4, 5, 6, 7, or 8 or more cars on any given workday are, respectively, 0.12, 0.19, 0.28, 0.24, 0.10, and 0.07, what is the probability that he will service at least 5 cars on his next day at work?

  19. Summary • Introduction to Probability • Additive Rules of Probability • Examples

  20. References • Probability and Statistics for Engineers and Scientists by Walpole

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