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Chapter 4, Part 1 Repeated Observations Independent Events The Multiplication Rule Conditional Probability 1 Repeated Observations / Trials • In this section, we discuss probabilities that arise when we observe: – The same random phenomenon more than once. – Two or more different random phenomena, each of which could effect the others. • We will use Classical Probability—all outcomes are equally likely. 2 Repeated Dice Rolling • Q: I roll my 20-sided die. What is the probability of rolling a 20? – Ans: 1/20 = 0.05 = 5% • Q: I roll my die a second time. What is the probability of rolling a 20? – Ans: The same as before: 1/20 = 0.05 = 5% • Q: I roll my die twice. What is the probability that both rolls are 20? 3 Repeated Dice Rolling • DIFFICULT(?) QUESTIONS: – I roll my die twice. You see that the first roll is 20. What is the probability that the second roll is 20? – I roll my die twice. You see that the second roll is 20. What is the probability that the first roll was 20? 4 Repeated Dice Rolling • Note the difference between: – Probability that both rolls were 20 (400 possible outcomes). – Probability that one roll was 20, given the result of the other roll (20 outcomes). • We usually assume that the two rolls “have no effect on one another,” but we don’t actually know this for certain. 5 Selection Without Replacement • Assume that my hat has 5 pink slips and 5 blue slips. • I pick one slip, and then pick a second slip without replacing the first. • Q: What is the probability that the first slip is blue? • Q: What is the probability that the second slip is blue? 6 Selection Without Replacement • The second probability depends on the result of the first selection. – First slip is Blue: There are 4 Blue, 5 Pink slips left. The probability that the second slip is Blue is 4/9. – First slip is Pink: There are 5 Blue, 4 Pink slips left. The probability that the second slip is Blue is 5/9. • This example illustrates what we will call a conditional probability. 7 Conditional Probability • The probability of one event occurring, assuming that another event has occurred is called a conditional probability. • Idea: Knowing/assuming that one event occurred might change the probability of a second event. • This commonly happens when we “select without replacement.” The first result has an effect on the second (and later) results. • Note that this didn’t happen with the multiple die rolls. The first result does not have an effect on the second (or so we assume). 8 Conditional Probability • We use the notation P(B|A), read as “the conditional probability of B, given A.” – Note that the order of events matters!!! • This means: We assume that event A has occurred, and use this knowledge to compute the (classical) probability of B. • P(B|A) and P(A|B) will usually be different numbers. Order of events matters!!! Conditional Probability Example • I have 5 blue and 5 pink slips in the hat. I choose two slips, without replacing the first. • P(2nd is blue | 1st is Pink) = ?? – Assuming the 1st is pink (note the order of events), the hat contains 5 blue, 4 pink. – The conditional probability is 5/9. • P(2nd is blue | 1st is Blue) = ?? – After the 1st selection, the hat contains 4 blue, 5 pink. – The conditional probability is 4/9. The Multiplication Rule • Notation (only in section 4-4): – We observe a random phenomenon twice. – “A and B” means that event A occurs on the first observation, and event B occurs on the second observation. – While this is technically a correct use of “A and B,” it requires that each “outcome” consist of two separate observations. This will probably confuse you, so… – I will use the notation “A then B” to emphasize that we are considering two observations. The Multiplication Rule • For any two events A, B. P(A then B) = P(A) x P(B|A) • Example: I choose two slips from the hat (without replacing the first). The probability that both slips are blue is: P(Blue then Blue) = (5/10)*(4/9) = 2/9. The Multiplication Rule • P(A then B) = P(A) x P(B|A) • The probability that the 1st is blue and the 2nd is pink? P(Blue then Pink) = (5/10)*(5/9) = 5/18. The Actual Definition of Conditional Probability • The conditional probability of B given A (in that order) is defined to be: P(A and B) P(B | A) P(A) • This is the standard use of “A and B” discussed last time. “A then B” is just a special case of this. 14 Classical Conditional Probability • For conditional probability, we assume that one event (A) occurred, and use this information to compute the probability of a second event (B). • If we make a single observation, and all outcomes are equally likely, there is a very simple method for computing conditional probability… 15 Classical Conditional Probability • For the result of a single observation, P(B|A) is given by: # of outcomes in “A and B” # of outcomes in A • In other words, we take the number of outcomes satisfying BOTH conditions, and divide by the number of outcomes satisfying the GIVEN condition. 16 Examples • I select a student from the class. Assume all outcomes are equally likely, compute: – P(Female | Row 1) – P(Male | Back Row) – P(Back Row | Male) – P(Back Row) Dependent Events • More often than not, knowing that one event occurred will change our expectations about a second event. • In terms of conditional probability, this means that P(B) ≠ P(B|A). • If this is the case, we say that the two events are dependent. Independent Events • But in some cases, knowing that one event occurred DOES NOT change our expectations about a second event. • In terms of conditional probability, this means that P(B) = P(B|A). If this is true, we say the events are independent. • An example is the 20-sided die: P(2nd is 20) = P(2nd is 20 | 1st is 20). Independent Events, Example • I have 5 blue, 5 pink slips in the hat. I choose two slips, REPLACING THE FIRST before drawing the second. Compute the following: – P(2nd is Blue) – P(2nd is Blue | 1st is Blue) • You should find these to be the same number, so these are independent events. The Multiplication Rule for Independent Events • Original multiplication rule: P(A then B) = P(A) x P(B|A) • If the events are independent, then P(B|A) is equal to P(B). The multiplication rule reduces to: P(A then B) = P(A) x P(B)