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# Week 8 Lecture Notes MATH 105-02

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This 5 page Class Notes was uploaded by Danielle Kelly on Friday March 6, 2015. The Class Notes belongs to MATH 105-02 at Washington State University taught by Spencer Payton in Spring2015. Since its upload, it has received 77 views. For similar materials see Exploring Mathematics in Mathematics (M) at Washington State University.

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Date Created: 03/06/15

Week 8 lecture notes Chapter 11 Section 2 Involving quotNotquot and quotOrquot Properties of Probability Let E be an event within some spaces S 1 0 s PE s 1 2 PQ 0 impossible event 3 PS certain event Example When a single fair die rolled nd the probability of each event S 123456 A 2 is rolled 1 P2 g B other than 2 is rolled E 5 PE g C 7 is rolled P7 0 impossible event D less than 7 is rolled 6 P gt 7 g 1 Complements nE nS nE39 nS nnot E nnotE nS nE nS Pnot E nS nS nS nS 1PE SEUE 39 nEUE39 E EI PS 1 PEUE ns nS quotElPEPEI1 1 nS nS Complements idea PE39 1 PE Example When a single card is drawn from a standard 52 card deck the probability that it will NOT be a kind is 4 52 4 4812 Pnot a kind 1Pking 1 5 2 5 2 5 252E Or nnotaking4812 nNot a King 52 4 48 Pnot a king nS 5 2B Example If 5 fair coins are tossed nd the probability of obtaining at least two heads 2532 possible outcomes nS 32 E at least two heads nE 325Co5C1 nS n E nE26 13 WE nS32 16 Example If one is randomly selected from 12345678910 nd the probability that it will be a Odd or multiple of 4 S 12310 A 13579 B 48 nAorBnAnB nAOB52 0l PA 0quot B nS nS 10 10 b Odd or multiple of 3 A 13579 C 369 A C 39 nAvCnAnB nA C53 2 MAO nS ns 10 10 March 4 2015 Pnot E 1PE Two events A and B they are mutually exclusive events if they have no outcomes in common Cant occur simultaneously are disjoint A n B0 Addition Rule of Probability PA or B PA PB PA and B nA BnAnB nA BnA nB nA B PA or B nS nS n5n5 nS PA PB PA n B If A and B are mutually exclusive events PA or B PA PB Example If a single card is drawn from a 52card deck what is the probability that It will be a spade or a red card Spadenred Q spades are black 13 26 393 Pspade or red Pspade Pred 5 2 5 2 Z red cards diamonds and hearts 3 suits out of 4 Example Amy plans to spend 16 hours on homework X of hours spend random variable X PX 05 10 20 40 10 OWU39lbUJNH 15 Find the probability that Amy will spend A B C D Fewer than 3 hours Pxlt3 P 1 or 2 P 1 or 2 P1 P2 P 05 P 10 15 More than two hours Pmore than 2 P2ltx P 3 or 4 or 5 or 6 P3 P4 P5 P6 P20 P40 P10 P15 085 Or Pmore than 2 1P1P2 1015 085 More than 1 but no more than 5 hours P1ltxlt5 P2 P3 P4 P5 10 20 40 10 080 Fewer than 5 hours Px lt 5 P1 P2 P3 P4 005 010 020 040 075 Or Px lt 5 1P 5 or 6 1 PM P6 1010 015 75 Example Find the probability that a single card drawn from a 52card deck will be a diamond or a face card PDiamonds or Face card Pdiamonds P face card P Diamond and face card PDiamonds or Face card 5quot 13 12 3 2211 52 525226 March 6 2015 Chapter 11 section 3 Conditional Probability The probability of event B computed on the assumption that event A has happened is called the conditional probability of B given A and is denoted PBA Example 5 12345678910 A single number is selected randomly from S Find the following probabilities give the events A the selected number is odd B the selected number is a multiple of 3 nB 3 a PB MS b PA n B13579 n 36939 nA Bi1 PA and B W10 c The selected is a multiple of 3 given that the number is odd nAnB2 PBA W E PBA quotAmmssnli nAnBquotlS i nA nS nS 1 AnB1PAnB nls nu PlAnBl nlArPlAnBlrim P pm pm 115 nAnBPAnB PBA nA 19A Two events A and B are called independent events if knowledge about the occurrence of one of them has no effect on the probability of the other That is if PBA PB PAB PA Multiplication Rule of Probability If A and B are any two events PA and B PA x PBA since PBA PBnA PA If A and B are independent then PA and B PA x PB since PBAPB

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