Intro to Probabiltiy theory notes for Elements Class.pptx

Example ref table 1 what is the probability that a

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Example: Ref. (Table 1) what is the probability that a person picked at a random from the 111 subjects will be a male (M) and be a person who has used cocaine 100 times or more during his lifetime (C)? Solution: The probability is written as P (M∩C) (M∩C) indicates the joint occurrence of conditions M and C. No. of subjects satisfying both of the desired conditions is found in Table-1 at the intersection of the column labeled M and the row labeled C i.e. 25 since the selection will be made from the total set of subjects, the denominator is 111. P(M∩C) = 25/111 = 0.2252
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33 Multiplication Rule A probability may be computed from other probability. For example, a joint probability may be computed as the product of an appropriate marginal probability and an appropriate conditional probability. This relationship is known as the multiplication rule of probability. We may state the multiplication rule in general terms as follows for any two events A & B: P(A ∩ B) = P(B) P(A|B), if P(B) ≠ 0 or P(A ∩ B) = P(A) P(B|A), if P(A) ≠ 0
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34 Conditional Probability The conditional probability of A given B is equal to the probability of A∩B divided by the probability of B, provided the probability of B is not zero. That is => P (A|B) = P (A ∩ B)/ P (B), P (B) ≠ 0 and same way P (B|A) = P (A ∩ B)/ P (A), P(A) ≠ 0 Example : Using the equation and the date of Table-1 to find the conditional probability, P(C|M). Solution: According to the equation P(C|M) = P(C∩M)/ P(M) Earlier we found P(C∩M) = P(M∩C) = 25/111 = 0.2252.We have also determined that P(M) = 75/111 =0.6757. Using these results we are able to compute P(C|M) = 0.2252/ .6757 = 0.3333
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35 Addition Rule: Given two events A and B, the probability that event A, or event B, or both occur is equal to the probability that event A occurs, plus the probability that event B occurs, minus the probability that the events occur simultaneously. The addition rule may be written P(AUB) = P(A) + P(B) – P(A∩B)
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Example: If we select a person at random from the 111 subjects represented in Table-1 what is the probability that this person will be a male (M) or will have used cocaine 100 times or more during his lifetime(C) or both? The probability is P(MUC).By the addition rule as expressed this probability may be written as P(MUC) = P(M) + P(C) - P(M∩C) We have already found that P(M)=75/111=0.6757, and P(M∩C) = 25/111 = 0.2252; P(C) = 34/111= 0.3063 Substituting these results in to the equation for P(MUC)= 0.6757+0.3063-0.2252 = 0.7568. 36
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37 Independence of Events Suppose that the probability of event A is the same regardless of whether or not B occurs. In this situation, P(A|B) = P(A). In such cases we say that A and B are independent events. The multiplication rule for two independent events, then may be written as: P(A∩B) = P(B)P(A); P(A) ≠ 0, P(B) ≠ 0 Thus we see that if two events are independent, the probability of their joint occurrence is equal to the product of the probabilities of their individual occurrences.
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