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SECTION 6.3

SECTION 6.3. GENERAL PROBABILITY RULES. General addition rule P(A or B) = P(A) + P(B) – P(A and B) Addition rule for disjoint events P(one or more of A, B, C) = P(A) + P(B) + P(C) Multiplication rule for independent events P(A and B) = P(A)P(B). Review of Previous Rules.

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SECTION 6.3

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  1. SECTION 6.3 GENERAL PROBABILITY RULES

  2. General addition rule P(A or B) = P(A) + P(B) – P(A and B) Addition rule for disjoint events P(one or more of A, B, C) = P(A) + P(B) + P(C) Multiplication rule for independent events P(A and B) = P(A)P(B) Review of Previous Rules

  3. Conditional Probability • Conditional probability – the probability of one event under the condition that we know another event. • The “|” can be interpreted as “given the information that”

  4. General addition rule • P(A or B) = P(A) + P(B) – P(A and B) • P(A U B) = P(A) + P(B) – P(A ∩ B) • General multiplication rule • P(A and B) = P(A)P(B given A) • P(A ∩ B) = P(A)P(B|A) • When P(A) > 0, • Testing for independence: • Two events A and B are independent if P(B|A) = P(B)

  5. TESTING FOR INDEPENDENCE • Two events A and B are independent if P(B|A) = P(B) Think back to your last quiz. When rolling a die and then flipping a coin, let event A be getting a 1 or 2 on the roll of the die. Let event B be getting an even number on the die. Are A and B independent? P(B|A) = (2/12)/(4/12) = ½ P(B) = 6/12 = ½ Therefore, A and B are independent.

  6. A BLUE REPRESENTS DESIGNATED AREA A B

  7. AC BLUE REPRESENTS DESIGNATED AREA A B

  8. B BLUE REPRESENTS DESIGNATED AREA A B

  9. BC BLUE REPRESENTS DESIGNATED AREA A B

  10. A∩B BLUE REPRESENTS DESIGNATED AREA A B

  11. (A∩B)C BLUE REPRESENTS DESIGNATED AREA A B

  12. AUB BLUE REPRESENTS DESIGNATED AREA A B

  13. (AUB)C BLUE REPRESENTS DESIGNATED AREA A B

  14. Taste In Music • Musical styles other than rock and pop are becoming more popular. A survey of college students find that 40% like country music, 30% like gospel music, and 10% like both. • What is the conditional probability that a student likes gospel music if we know that he/she likes country music? Conditional Probability P(G|C) = 0.1/0.4=0.25 C G 30% 10% 20%

  15. Taste In Music (cont.) • Musical styles other than rock and pop are becoming more popular. A survey of college students find that 40% like country music, 30% like gospel music, and 10% like both. • What is the conditional probability that a student who does not like country music likes gospel music? Conditional Probability P(G|CC) = 0.2/0.6=1/3 C G 30% 10% 20%

  16. Venn Diagram PracticeRESTAURANT T C 15 12 9 11 7 13 10 23 Q

  17. RESTAURANT • 1. (CUTUQ)C P(CUTUQ)C = 23/100 • 2. (C∩T∩Q) P (C∩T∩Q) = 11/100 • 3. (CUQ∩TC) P(CUQ∩TC) = 29/100 • 4. (Q) P(Q) = 41/100 • 5. (T∩Q)U(Q∩C)U(C∩T) • P(T∩Q)U(Q∩C)U(C∩T) = 46/100 • 6. (T∩Q∩CC) P(T∩Q∩CC) = 13/100 • 7. (T∩C) P (T∩C) = 26/100

  18. Venn Diagram PracticeCARTOONS T A 23 17 73 9 31 14 11 22 P

  19. Venn Diagram PracticeCONCERT D D P 11 21 27 10 13 15 35 18 G G

  20. Venn Diagram PracticeSTAR TREK T D 23 17 73 9 31 14 11 22 V

  21. Venn Diagram PracticeMYTHOLOGY L H 5 12 18 3 6 7 24 25 R

  22. Venn Diagram PracticePOLLUTANTS P C 122 137 101 28 72 152 211 177 S

  23. Venn Diagram PracticeTENNIS S B 40 52 35 30 10 8 5 20 F

  24. Venn Diagram PracticeTENNIS TOURNAMENTS U W 30 40 50 10 20 15 30 5 A

  25. Venn Diagram PracticeLANGUAGES F G 8 29 20 4 16 12 92 27 S

  26. Nobel Prize Winners • If a laureate is selected at random, what is the probability that: • his or her award is in chemistry? • the award was won by someone working in the US? • the awardee was working in the US, given the award was for phys./med? • the award was for phys./med., given that the awardee was working in the US? 119/399 ≈ 0.2982 215/399 ≈ 0.5388 90/142 ≈ 0.6338 90/215 ≈ 0.4186

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