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.

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Presentation transcript:

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 –P(A and B) = P(A)P(B)

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.)

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)

Disjoint, yes? P(A or B) = P(A) + P(B) AB

Disjoint? No! Now P(A or B) = P(A) + P(B) – P(A  B) AB 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! AB A  B !!

So—In general: P(A or B) = P(A) + P(B) – P(A  B) AB

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?

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) = =.5 P(D  C) = =.5

Dartmouth/Cornell Or, adding the areas, P(Dartmouth or Cornell) = =