MATH112-WEEK 7-LESSON 12-Laws of Probabilities.pdf

STATISTICAL ANALYSIS WITH SOFTWARE APPLICATION Laws of Probability MODULE 2-WEEK 7-LESSON 12 Excellence and Relevance REMELYN L. ASAHID-CH...

STATISTICAL ANALYSIS WITH SOFTWARE APPLICATION Laws of Probability MODULE 2-WEEK 7-LESSON 12 Excellence and Relevance REMELYN L. ASAHID-CHENG What are the 4 Laws of Probability? Probability deals with the occurrence of a random event. Basic 4 laws of probability are given below: 1. Addition rule: P(A or B) = P(A) + P(B) – P(A and B) 2. Multiplication rule: P(A and B) = P(A). P(B/A) 3. The sum of the probabilities of all possible outcomes = 1 4. Complementary law: P(not A) = 1 – P(A) Excellence and Relevance  Addition Law The addition law of probability (sometimes referred to as the addition rule or sum rule), states that the probability that A or B will occur is the sum of the probabilities that A will happen and that B will happen, minus the probability that both A and B will happen. The addition rule is summarized by the formula: P(A∪B)=P(A)+P(B)−P(A∩B) Excellence and Relevance Addition Law Excellence and Relevance Addition Law Given P(A) =.10, P(B) =.12, P(C) =.21, P(A ∩ C) =.05, and P(B ∩ C) =.03, solve the following. (Round your answers to 2 decimal places, e.g. 0.21.) Solution: P(A) =.10, P(B) =.12, P(C) =.21, P(A ∩ C) =.05 P(B ∩ C) =.03 a) P(A∪C) = b) P(B∪C) = c) If A, B mutually exclusive, P(A∪B) = P(A) + P(B) = Excellence and Relevance Addition Law Use the values in the joint probability table to solve the equations given. Solution: Excellence and Relevance Addition Law According to a survey conducted by Netpop Research, 65% of new car buyers use online search engines as part of their car-buying experience. Another study reported that 11% of new car buyers skip the test drive. Suppose 7% of new car buyers use online search engines as part of their car-buying experience and skip the test drive. If a new car buyer is randomly selected, what is the probability that: Solution: Excellence and Relevance The Addition Rule Use the values in the cross-tabulation table to solve the equations given. Excellence and Relevance Addition Law Addition Rule for Disjoint Events Suppose A and B are disjoint, their intersection is empty. Then the probability of their intersection is zero. In symbols: P(A∩B)=0. The addition law then simplifies to: P(A∪B) = P(A)+P(B) when A∩B = ∅ The symbol ∅ represents the empty set, which indicates that in this case A and B do not have any elements in common (they do not overlap). Excellence and Relevance The Addition Rule According to Nielsen Media Research, approximately 86% of all U.S. households have High-definition television (HDTV). In addition, 49% of all U.S. households own Digital Video Recorders (DVR). Suppose 40% of all U.S. households have HDTV and have DVR. A U.S. household is randomly selected. a. What is the probability that the household has HDTV or has DVR? b. What is the probability that the household does not have HDTV or does have DVR? c. What is the probability that the household does have HDTV or does not have DVR? d. What is the probability that the household does not have HDTV or does not have DVR? Excellence and Relevance The Multiplication Rule In probability theory, the Multiplication Rule states that the probability that A and B occur is equal to the probability that A occurs times the conditional probability that B occurs, given that we know A has already occurred. This rule can be written: P(A∩B)= P(B)⋅P(A|B) Switching the role of A and B, we can also write the rule as: P(B∩A)= P(A)⋅P(B|A) We obtain the general multiplication rule by multiplying both sides of the definition of conditional probability by the denominator. That is, in the equation P(A|B)=P(A∩B)/P(B), if we multiply both sides by P(B), we obtain the Multiplication Rule. The rule is useful when we know both P(B) and P(A|B), or both P(A) and P(B|A). Excellence and Relevance The Multiplication Rule Excellence and Relevance The Multiplication Rule Use the values in the cross-tabulation table to solve the equations given. A. P(A∩E) = b. P(D∩B) = c. P(D∩E) = d. P(A∩B) = Excellence and Relevance The Multiplication Rule If A and B are two independent events in a probability experiment, then the probability that both events occur simultaneously is: P(A and B)=P(A)⋅P(B) In case of dependent events, the probability that both events occur simultaneously is: P(A and B)=P(B)⋅P(A|B) (The notation P(B | A) means "the probability of B, given that A has happened.") Excellence and Relevance Excellence and Relevance The Multiplication Rule a. A batch of 30 parts contains six defects. If two parts are drawn randomly one at a time without replacement, what is the probability that both parts are defective? b. If this experiment is repeated, with replacement, what is the probability that both parts are defective? Solution: Excellence and Relevance The Complementary Rule Two events are said to be complementary when one event occurs if and only if the other does not. The probabilities of two complimentary events add up to 1. For example, rolling a 5 or greater and rolling a 4 or less on a die are complementary events, because a roll is 5 or greater if and only if it is not 4 or less. a. For example, rolling a 5 or greater and rolling a 4 or less on a die are complementary events, because a roll is 5 or greater if and only if it is not 4 or less. The probability of rolling a 5 or greater is 2/6 = 1/3, and the probability of rolling a 4 or less is 4/6 = 2/3. Thus, the total of their probabilities is 1/3+ 2/3 = 1. b. Another example, if the probability of an event is 3/8, what is the probability of its complement? The probability of its complement is 1 – 3/8 = 8/8 - 3/8 = 5/8. Excellence and Relevance  End of LESSON 12 Excellence and Relevance REMELYN L. ASAHID-CHENG