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General Probability Rules 6.3

General Probability Rules 6.3. Mathematical Notation for Probability Rules. If A = some event, then 0 ≤ P(A) ≤ 1 If S = sample space, then P(S) = 1 P(A c ) = 1 - P(A) If A and B are disjoint: P(A or B) = P(A)+P(B) If A and B are independent: P(A and B) = P(A)P(B). Union.

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General Probability Rules 6.3

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  1. General Probability Rules 6.3

  2. Mathematical Notation for Probability Rules If A = some event, then 0 ≤ P(A) ≤ 1 If S = sample space, then P(S) = 1 P(Ac) = 1 - P(A) If A and B are disjoint: P(A or B) = P(A)+P(B) If A and B are independent: P(A and B) = P(A)P(B)

  3. Union of any collection of events is the event that at least one of the collections occurs. “OR” Example: Suppose S = {deck of 52 cards} and A = {spades} and B = {hearts}. Then A U B (read: A union B ) or (read: A or B) = {spades, hearts}. So drawing any spade or drawing any heart is a member of A U B.

  4. Disjoint events S B A

  5. Addition Rule for Disjoint Events If events A, B, C,..., Z are disjoint, then P(A or B or ... Z) = P(A) + P(B) + ... + P(Z) Example: Using our first card example above, find P(A U B). Solution: A U B = {spades, hearts} = 26 possibilities. P(A or B) = 13/52 + 13/52 = 26/52 = 1/2. Remember: P(A or B) = P(A U B).

  6. But what if our events aren’t disjoint Notice that if we did P(A) + P(B) this time, we’d be counting the intersection of A and B twice! So our formula logically changes to…

  7. FOR ANY EVENTS A and B Note: This formula works for disjoint events as well since for disjoint events is zero! (Draw the Venn Diagram)

  8. Tesia and Courtney are anxiously awaiting word on whether they have been made partners in a law firm. Tesia’s probability of success is 0.7 and Courtney thinks hers is 0.5. The probability they are both promoted is 0.3. Find: a. Probability at least 1 person is promoted? b. Probability that Tesia is promoted and Courtney isn’t? c. Vice versa of part b.

  9. The multiplication rule gets messy when we talk about more than two events. Suppose I wanted the P(A and B and C). P(A)P(B|A)P(C|A and B) A tree diagram is helpful.

  10. 6.63 (tree) 6.64 (tree)

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