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Notes nov 1 stats

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Notes nov 1 stats Notes nov 1 stats Presentation Transcript

  • November 1st Stt 200 Notes By: Zach Hamden
  • Function P • 0 is greater than P, P is less than 1 • P(S)= 1 • Example • Probability for 3 different things, all probabilities added up must equal 1. • Idea of property distrubtion 2
  • EXAMPLES 1. THREE STUDENTS A, B AND C ARE IN A SWIMMING RACE. A AND B HAVE THE SAME PROBABILITY OF WINNING AND EACH IS TWICE AS LIKELY TO WIN AS C. FIND THE PROBABILITY THAT B OR C WINS. 2. Sample Space = A, B, C 3. Probability of C equals X, Probabilty of A and B equals X 4. Probability of P(A) + P(B) + P (C)= 1 5. 2x + 2x + 2x = 1 6. 5x=1, or x= 1/5 7. P(C) = 1/5 P(A) 2(1/5)=2/5 P(B) 2(1/5)= 2/5 8. Not Done...P (B or (Or means add) C)= 2/5 + 1/5= 3/5 9. Not Formulas, Properties will not get them on 3 formula sheet.
  • EVENTS • AN EVENT IS A SUBSET OF A SAMPLE SPACE, THAT IS, A COLLECTION OF OUTCOMES FROM THE SAMPLE SPACE. • EVENTS ARE DENOTED BY UPPER CASE LETTERS, FOR EXAMPLE, A, B, C, D. • LET E BE AN EVENT. THEN THE PROBABILITY OF E, DENOTED P(E), IS GIVEN BY • Example: Toss Die sample space is (1 thru 6) • Event e= even numbers (2, 4, 6) • Event o= odd numbers (1, 3, 5) • Find probabilities of all the outcomes in event e, find probabilities of each outcome and add them. 4
  • FOR ANY EVENT E, 0 < P(E) < 1 • COMPUTATIONAL FORMULA • LET E BE ANY EVENT AND S THE SAMPLE SPACE. THE PROBABILITY OF E, DENOTED P(E) IS COMPUTED AS • Example: Sample Space = 2 blue balls, 5 green balls, 4 red balls • What is Chance you pick green ball • P (G (green balls))= n (G)/n (S(all balls))= 5/11 5
  • EXAMPLES 1. ONE CARD IS SELECTED AT RANDOM FROM 50 CARDS NUMBERED 1 TO 50. FIND THE PROBABILITY THAT THE NUMBER ON THE CARD IS (I) DIVISIBLE BY 5, (II) PRIME, (III) ENDS IN THE DIGIT 2. 2. Need set of all numbers divisible by 5 (list numbers), D5 (5, 10, 15, 20, 25, 30, 35, 40 ,45, 50) 3. P(D5)= 10/50, (number of cards divisible by 5)/ (Total Sample, Number of cards) 4. 2nd Part : Prime Number, Need to list prime numbers, P(r)= (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47) 5. P (P(r))= 15/50= 3/10 (number of prime)/ (Total Sample, Number of cards) 6
  • Null Event • Null Event- Event has no chance of happening, probability is 0. P(E)= 0 • Certain Event- Event will occur, probability is 1. P(E)= 1 • 7
  • COMBINATION OF EVENTS • INTERSECTION OF EVENTS THE INTERSECTION OF TWO EVENTS A AND B, DENOTED, Overlapping Events A Intersection B -IS THE EVENT CONTAINING ALL ELEMENTS(OUTCOMES) THAT ARE COMMON TO A AND B. -Venn Diagrams Portray Events and intersection Events A and B A BB A and B 8
  • COMPLEMENT OF AN EVENT • THE COMPLEMENT OF AN EVENT A WITH RESPECT TO S IS THE SUBSET OF ALL ELEMENTS(OUTCOMES) THAT ARE NOT IN A. • Know Probability of A complement or A you know the other one. • NOTATION: 9
  • MUTUALLY EXCLUSIVE(DISJOINT) EVENTS • TWO EVENTS A AND B ARE MUTUALLY EXCLUSIVE(DISJOINT) IF THAT IS, A AND B HAVE NO OUTCOMES IN COMMON. Have No overlap, Venn Diagram Ex) A B IF A AND B ARE DISJOINT(MUTUALLY EXCLUSIVE), 10
  • ADDITION RULE • IF A AND B ARE MUTUALLY EXCLUSIVE EVENTS, THEN, (No overlaps or nothing in common) GENERAL ADDITION RULE IF A AND B ARE ANY TWO EVENTS, THEN (Must subtract probability of intersection, because it’s considered in both event sets, already accounted for in probability of events) 11