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Probability Rules! When two events A and B are disjoint, we have P (A or B) = P (A) + P (B) However, when the events are not disjoint, this addition rule will double count the probability of both A and B occurring. Thus, we need the General Addition Rule: P (A or B) = P (A) + P (B) – P (A and B) The following Venn diagram shows a situation in which we would use the general addition rule: A check of dorm rooms on a college campus revealed that 54% had TVs, 64% had refrigerators, and 43% had both a TV and a refrigerator. What’s the probability that a randomly selected dorm room has a refrigerator but no TV? either a refrigerator or a TV? neither refrigerator nor TV? When we want the probability of an event from a conditional distribution, we write P (B|A) and pronounce it “the probability of B given A.” A probability that takes into account a given condition is called a conditional probability. To find the probability of the event B given the event A, we restrict our attention to the outcomes in A. We then find in what fraction of those outcomes B also occurred. P(B| A) P(A and B) P(A) Draw a card at random from a deck of 52 cards. What’s the probability that the card is a spade, given that it is black? the card is a king, given that it is red? the card is a king, given that is a face card? the card is red, given that it is a diamond? Independence of two events means that the outcome of one event does not influence the probability of the other. Formally, events A and B are independent whenever P (B|A) = P (B) , or equivalently, whenever P (A|B) = P (A). Disjoint events cannot be independent: ◦ Since we know that disjoint events have no outcomes in common, knowing that one occurred means the other didn’t. ◦ Thus, the probability of the second occurring changed based on our knowledge that the first occurred. ◦ It follows, then, that the two events are not independent. When two events A and B are independent, P (A and B) = P (A) x P (B) However, when our events are not independent, the above rule does not work. Thus, we need the General Multiplication Rule: P (A and B) = P (A) x P (B|A) or P (A and B) = P (B) x P (A|B) Sampling without replacement means that once one object is drawn it doesn’t go back into the pool. ◦ We often sample without replacement, which doesn’t matter too much when we are dealing with a large population. ◦ However, when drawing from a small population, we need to take note and adjust probabilities accordingly. Drawing without replacement is just another instance of working with conditional probabilities. Draw two cards at random from a deck of 52 cards. What’s the probability that both cards are black? at least one card is red? the first card is an ace? A tree diagram helps us think through conditional probabilities by showing sequences of events as paths that look like branches of a tree. Examples: ◦ Toss a coin 3 times. ◦ Suppose a person aged 20 has about an 80% chance of being alive at age 65. Suppose that three people aged 20 are selected at random. What’s the probability that exactly two people will be alive at age 65? Solution Suppose we want to know P (A|B), but we know only P (A), P (B), and P (B|A). We also know P (A and B), since P (A and B) = P (A) x P (B|A) From this information, we can find P (A|B): P(A|B) P(A and B) P(B) When we reverse the probability from the conditional probability that you’re originally given, We obtain Bayes’s Rule. P A | B P B P B | A C C P A | B P B P A | B P B Page 404 – 407 Problem # 1, 3, 5, 7, 9, 11, 15, 17, 19, 23, 27, 29, 33, 35, 43, 45.