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Probability 2

Probability 2. Area of a Square. 100%. Area of Green Square (X). X = 25%. Area of NOT-X. ~X = 75%. Formula. Area(~X) = 1 – Area(X). Area of X. X = 25%. Area of A. X = 25%. A = 25%. Area of X v A. X = 25%. 50%. A = 25%. Formula. If X and A are non-overlapping, then

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Probability 2

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  1. Probability 2

  2. Area of a Square 100%

  3. Area of Green Square (X) X = 25%

  4. Area of NOT-X ~X = 75%

  5. Formula Area(~X) = 1 – Area(X)

  6. Area of X X = 25%

  7. Area of A X = 25% A = 25%

  8. Area of X v A X = 25% 50% A = 25%

  9. Formula If X and A are non-overlapping, then Area(X v A) = Area(X) + Area(A)

  10. Area of Y Y = 50%

  11. Area of Z Z = 50%

  12. Area of Y or Z Y v Z = 75%

  13. Formula Area(Y v Z) = Area(Y) + Area(Z) – Area(Y & Z)

  14. Area of Y & Z Y & Z = 25%

  15. Independence Y and Z are independent: knowing that a point is in Y does not increase the probability that it’s in Z, because half of the points in Y are in Z and half are not.

  16. Formula If Y and Z are independent, then Area(Y & Z) = Area(Y) x Area(Z)

  17. Area of Z Z = 50%

  18. Area of B 50%

  19. Area of B v Z 62.5%

  20. Area(Z & B) Z & B = 37.5%

  21. Correlated Areas B and Z are not independent. 75% of the points in Z are also in B. If you know that a point is in Z, then it is a good guess that it’s in B too.

  22. Formula Area(B & Z) = Area(B) x Area(Z/ B) This is the percentage of B that is in Z: 75%

  23. Area(Z & B) Z & B = 37.5%

  24. Area(Z & B) Z & B = 37.5% Z = 50%

  25. Conditional Areas Area(Z/ B) = Area(Z & B) ÷ Area (B) = 37.5% ÷ 50% = 75%

  26. Formula Area(B & Z) = Area(B) x Area(Z/ B) = 50% x 75% = 37.5%

  27. Area(Z & B) Z & B = 37.5%

  28. Rules Area(~P) = 1 – Area(P) Area(P v Q) = Area(P) + Area(Q) – Area(P & Q) Area(P v Q) = Area(P) + Area(Q) for non-overlapping P and Q Area(P & Q) = Area(P) x Area(Q/ P) Area(P & Q) = Area(P) x Area(Q) for independent P and Q

  29. Rules Pr(~φ) =1 – Pr(φ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) when φ and ψ are mutually exclusive Pr(φ & ψ) = Pr(φ) x Pr(ψ/ φ) Pr(φ & ψ) = Pr(φ) x Pr(ψ) when φ and ψ are independent

  30. In-Class Exercises

  31. 1. Pr(P) = 1/2, Pr(Q) = 1/2, Pr(P & Q) = 1/8, what is Pr(P v Q)? 2. Pr(R) = 1/2, Pr(S) = 1/4, Pr(R v S) = 3/4, what is Pr(R & S)? 3. Pr(U) = 1/2, Pr(T) = 3/4, Pr(U & ~T) = 1/8, what is Pr(U v ~T)?

  32. Known: Pr(P) = 1/2, Known: Pr(Q) = 1/2, Known: Pr(P & Q) = 1/8 Unknown: Pr(P v Q)

  33. Rules Pr(~φ) =1 – Pr(φ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) when φ and ψ are mutually exclusive Pr(φ & ψ) = Pr(φ) x Pr(ψ/ φ) Pr(φ & ψ) = Pr(φ) x Pr(ψ) when φ and ψ are independent

  34. Pr(P v Q) Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) Pr(P v Q) = Pr(P) + Pr(Q) – Pr(P & Q) = 1/2 + Pr(Q) – Pr(P & Q) = 1/2 + 1/2 – Pr(P & Q) = 1/2 + 1/2 – 1/8 = 7/8 Known: Pr(P) = 1/2, Known: Pr(Q) = 1/2, Known: Pr(P & Q) = 1/8

  35. Known: Pr(R) = 1/2 Known: Pr(S) = 1/4 Known: Pr(R v S) = 3/4 Unknown: Pr(R & S)?

  36. Not Helpful: More than One Unknown Pr(~φ) =1 – Pr(φ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) when φ and ψ are mutually exclusive Pr(φ & ψ) = Pr(φ) x Pr(ψ/ φ) Pr(φ & ψ) = Pr(φ) x Pr(ψ) when φ and ψ are independent

  37. This Is What You Want Pr(~φ) =1 – Pr(φ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) when φ and ψ are mutually exclusive Pr(φ & ψ) = Pr(φ) x Pr(ψ/ φ) Pr(φ & ψ) = Pr(φ) x Pr(ψ) when φ and ψ are independent

  38. Pr(R & S) Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) Pr(R v S) = Pr(R) + Pr(S) – Pr(R & S) 3/4 = Pr(R) + Pr(S) – Pr(R & S) 3/4 = 1/2 + Pr(S) – Pr(R & S) 3/4 = 1/2 + 1/4 – Pr(R & S) 3/4 = 3/4 – Pr(R & S) • Known: Pr(R) = 1/2 • Known: Pr(S) = 1/4 • Known: Pr(R v S) = 3/4

  39. Known: Pr(U) = 1/2 Known: Pr(T) = 3/4 Known: Pr(U & ~T) = 1/8 Unknown: Pr(U v ~T)?

  40. Rules Pr(~φ) =1 – Pr(φ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) when φ and ψ are mutually exclusive Pr(φ & ψ) = Pr(φ) x Pr(ψ/ φ) Pr(φ & ψ) = Pr(φ) x Pr(ψ) when φ and ψ are independent

  41. Pr(~T) Pr(~φ) =1 – Pr(φ) Pr(~T) = 1 – Pr(T) = 1 – 3/4 = 1/4 • Known: Pr(U) = 1/2 • Known: Pr(T) = 3/4 • Known: Pr(U & ~T) = 1/8 • Known: Pr(~T) = 1/4

  42. Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) • Known: Pr(U) = 1/2 • Known: Pr(T) = 3/4 • Known: Pr(U & ~T) = 1/8 • Known: Pr(~T) = 1/4

  43. Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) • Known: Pr(U) = 1/2 • Known: Pr(T) = 3/4 • Known: Pr(U & ~T) = 1/8 • Known: Pr(~T) = 1/4

  44. Pr(U v ψ) = Pr(U) + Pr(ψ) – Pr(U & ψ) • Known: Pr(U) = 1/2 • Known: Pr(T) = 3/4 • Known: Pr(U & ~T) = 1/8 • Known: Pr(~T) = 1/4

  45. Pr(U v ψ) = Pr(U) + Pr(ψ) – Pr(U & ψ) • Known: Pr(U) = 1/2 • Known: Pr(T) = 3/4 • Known: Pr(U & ~T) = 1/8 • Known: Pr(~T) = 1/4

  46. Pr(U v ~T) = Pr(U) + Pr(~T) – Pr(U & ~T) • Known: Pr(U) = 1/2 • Known: Pr(T) = 3/4 • Known: Pr(U & ~T) = 1/8 • Known: Pr(~T) = 1/4

  47. Pr(U v ~T) Pr(U v ~T) = Pr(U) + Pr(~T) – Pr(U & ~T) = 1/2 + Pr(~T) – Pr(U & ~T) = 1/2 + 1/4 – Pr(U & ~T) = 1/2 + 1/4 – 1/8 = 5/8 • Known: Pr(U) = 1/2 • Known: Pr(T) = 3/4 • Known: Pr(U & ~T) = 1/8 • Known: Pr(~T) = 1/4

  48. More Exercises 4. Suppose I flip a fair coin three times in a row. What is the probability that it lands heads all three times? 5. Suppose I flip a fair coin four times in a row. What is the probability that it does not land heads on any of the flips?

  49. Problem #4 4. Suppose I flip a fair coin three times in a row. What is the probability that it lands heads all three times? Known: Pr(F) = 1/2 Known: Pr(S) = 1/2 Known: Pr(T) = 1/2 Unknown: Pr((F & S) & T)

  50. Rules Pr(~φ) =1 – Pr(φ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) – Pr(φ & ψ) Pr(φ v ψ) = Pr(φ) + Pr(ψ) when φ and ψ are mutually exclusive Pr(φ & ψ) = Pr(φ) x Pr(ψ/ φ) Pr(φ & ψ) = Pr(φ) x Pr(ψ)when φ and ψ are independent

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