# Pa 15180 00833 pnot a 165 180 09167 complementary

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P(A) = 15/180 = 0.0833 P(not A ) = 165 / 180 = 0.9167
Complementary events: For any given event, its complement contains all the points that are not in the event. There are two rules with complementary events: Complementary events are always mutually exclusive events. The probability of complementary events always adds to 1. 4.2 Conditional Probability Conditional probability: determining the probability of an event occurring, given another event has already occurred. EXAMPLE 5: Suppose we pick a tasting at random. What is the probability that our randomly selected tasting has a “good” quality rating? IE . Anything else |. A = event of interest |. A e compliment pf A
EXAMPLE 6: Suppose we knew that the randomly selected tasting was a high priced beer. Then: How many tastings are known to be from high priced beers?
Of these high priced beers, how many of them have a good quality rating?
What is the probability that a randomly selected tasting has a good quality rating, knowing that it is a high priced beer? B :
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Independent Events : Two events are independent if the occurrence of one event has no effect on the probability of the occurrence of the other event EXAMPLE 7: We may be interested in knowing if high priced beers are more likely to receive a good quality rating, or if these two events are independent . We can test for independence by examining the conditional probabilities: First, determine the probability of a randomly selected tasting having a good quality rating: Second, determine the conditional probability of a randomly selected tasting have a good quality rating, given that it is known to be a high priced beer: Finally, compare these two probabilities: What happened? B changed the probability of A Mathematically, a Test for Independence can be expressed as: Independent if p(A) stays the same even if B happens first P ( A | B ) = P ( A ) A second Test for Independence (less intuitive, but still true) is: If two events ( A and B ) are known to be independent, then P ( A B ) = P ( A ) × P ( B ) is true. The mathematical tests can be applied in reverse: 34 A = good quality rating , p ( A ) = 36/180 =0.20 A = good quality rating |. P(A|| B ) = 15/60 =0.25 B = high price | P (A ) =/P(A|B) If P(A) = P(A|B) P(B) = P (B|A) P(A B ) = P(A) * p(B) Then A + B are independent + vice versa