Combining Probabilities Unit 7B Combining Probabilities Ms. Young

And Probability: Independent Events Two events are independent if the outcome of one does not affect the probability of the other event. If two independent events A and B have individual probabilities P(A) and P(B), the probability that A and B occur together is P(A and B) = P(A) • P(B). This principle can be extended to any number of independent events. Ms. Young

And Probability: Dependent Events Two events are dependent if the outcome of one affects the probability of the other event. The probability that dependent events A and B occur together is P(A and B) = P(A) • P(B given A) where P(B given A) is the probability of event B given the occurrence of event A. This principle can be extended to any number of dependent events. Help students understand that dependent events are those whose outcomes can be influenced by prior events. Practice makes perfect here. Ms. Young

Either/Or Probabilities: Non-Overlapping Events Two events are non-overlapping if they cannot occur together, like the outcome of a coin toss, as shown to the right. For non-overlapping events A and B, the probability that either A or B occurs is shown below. P(A or B) = P(A) + P(B) This principle can be extended to any number of non-overlapping events. Ms. Young

Either/Or Probabilities: Overlapping Events Two events are overlapping if they can occur together, like the outcome of picking a queen or a club, as shown to the right. For overlapping events A and B, the probability that either A or B occurs is shown below. P(A or B) = P(A) + P(B) – P(A and B) This principle can be extended to any number of overlapping events. Ms. Young

Examples What is the probability of rolling either a 3 or a 4 on a single six-sided die? These are non-overlapping events. P(A or B) = P(A) + P(B) P(3 or 4) = P(3) + P(4) = 1/6 + 1/6 = 2/6 = 1/3 What is the probability that in a standard shuffled deck of cards you will draw a 5 or a spade? These are overlapping events. P(A or B) = P(A) + P(B) – P(A and B) P(5 or spade) = P(5) + P(spade) – P(5 and spade) = 4/52 + 13/52 – 1/52 = 16/52 = 4/13 Here the focus should be on determining if there is overlap in event outcomes. Ms. Young

The At Least Once Rule (For Independent Events) Suppose the probability of an event A occurring in one trial is P(A). If all trials are independent, the probability that event A occurs at least once in n trials is shown below. P(at least one event A in n trials) = 1 – P(not event A in n trials) = 1 – [P(not A in one trial)]n This particular example can be verified quite easily with a look at the sample space of four children. Ask other similar questions to assess students’ understanding of the rule. Ms. Young