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6.3: General Probability Rules

6.3: General Probability Rules. The Rules So Far. 1. Probabilities are between 0 and 1 for any event A 2. The sum of all probs for a given sample space is 1 3. P(A c ) = 1 – P(A) 4. Addition Rule for Disjoint events P(A or B) = P(A) + P(B) 5. Multiplication Rule for Independent Events

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6.3: General Probability Rules

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  1. 6.3: General Probability Rules

  2. The Rules So Far • 1. Probabilities are between 0 and 1 for any event A • 2. The sum of all probs for a given sample space is 1 • 3. P(Ac) = 1 – P(A) • 4. Addition Rule for Disjoint events • P(A or B) = P(A) + P(B) • 5. Multiplication Rule for Independent Events • P(A and B) = P(A)P(B)

  3. Union: Definition and Rule • The Union of a collection of events is the event that at least one of the collection of events occurs • The Addition Rule for Disjoint Events: P(At least one occurrence happening from the set of events) = P(A) + P(B) + P(C) + …AS LONG AS THE EVENTS ARE DISJOINT!! (Sorry. Didn’t mean to shout.)

  4. The Addition Rule for Any two events, Disjoint or not • P(A or B) = P(A) + P(B) – P(A and B) or • P(A  B) = P(A) + P(B) – P(A  B)

  5. Disjoint, yes? • P(A or B) = P(A) + P(B) A B

  6. A B Disjoint? No! • Now P(A or B) = P(A) + P(B) – P(A  B) A  B !!

  7. A B • Why subtract P(A  B)? That overlap area (now orange) is covering up another area just like it in green. To gauge the true total area properly, we must throw one of them away! A  B !!

  8. A B So—In general: • P(A or B) = P(A) + P(B) – P(A  B)

  9. Example: Dartmouth/Cornell • George believes he has a .4 change of being accepted at Dartmouth, and a .3 chance of being accepted at Cornell. • Furthermore, he thinks he has a .2 chance of being accepted at both. • What is the probability of being accepted at either one? Dartmouth or Cornell?

  10. Dartmouth/Cornell • P(D or C) = P(D) + P(C) – P(D and C) • P(D  C) = P(D) + P(C) – P(D  C) P(D  C) = .4 + .3 - .2 = .5

  11. .1 Dartmouth/Cornell • Or, adding the areas, P(Dartmouth or Cornell) = .2 + .2 + .1 = .5 .2 .2 .

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