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Sometimes to calculate the probability of an event directly is tedious, while it is easier to obtain the probability of the complement of this event. Example 14. Probability of Employee Selection A Congresswoman wants to hire four assistants from eight candidates. Five candidates are men, and three are women. If every combination of candidates is equally likely to be chosen, what is the probability that at least one women will be hired? Example 15. Born the Same Day? (A great question for a party!) There are 29 students in class Econ390. What is the probability that at least two students have the same birthday? Conditional Probability Consider a pair of events A and B. Suppose we are concerned about the probability of A given that B has occurred. The idea is that the probability of an event is often affected by knowledge of circumstances. For example, if we randomly select someone from the general population, the probability of getting a male is 0.5, but if we know that the selected person frequently changes TV channels with a remote control, what is the probability in this case? Conditional Probability of A given B is denoted as P(A l B) and defined as P(A l B) = P (A ∩ B) / P(B) (Assumes P(B)≠0) Similarly, P(B l A) = P (A ∩ B) / P(A) (Assumes P(A)≠0) Use relative frequencies to help us understand conditional probability. From the definition of condition probability, we can obtain the Multiplication Rule of Probability P (A ∩ B) = P(B l A) P(A) Also, P (A ∩ B) = P(A l B) P(B) Example 1. Product Choice A hamburger chain found that 75% of all customers use mustard, 80% use ketchup, and 65% use both. If you are given the information that a randomly selected customer uses ketchup, what is the probability that she/he also uses mustard? Statistical Independence Event A and event B are said to be statistically independent if and only if P(B | A) = P(B) or P(A | B ) =P(A) which means the probability of B occurring doesn’t change even after given the information that A has occurred. If event A and event B are statistically independent, the multiplication Rule is P (A ∩ B) = P(B | A) P(A) = P(B)P(A) There are three ways to test if event A and B are statistically independent. Example 2. Probability of College Degrees Suppose that 48% of all bachelor degrees in a particular country are obtained by women and that 17% of all bachelor degrees are in business. Also, 6% of all bachelor degrees go to women majoring in business. Are the events “Bachelor degree holder is a women” and “Bachelor degree is in business” statistically independent? Statistically Independent Events ≠ Disjoint Events A and B independent means the information of A occurs or not does not affect the probability of B occurring If A and B are disjoint, the information of A occurring indicates that B does not occur. So if A and B are disjoint, they must be statistically dependent! If A and B are disjoint, P (A ∩ B) =0 If A and B are independent, P (A ∩ B) =P(A)P(B) Applying Multiplication Rule Example 3. Foul-ups A mail-order firm considers three possible foul-ups in filling an order: A: The wrong item is sent. B: The item is lost in transit. C: The item is damaged in transit. Assume that event A is independent of both B and C, and that events B and C are mutually exclusive. The individual event probabilities are P(A)=0.02, P(B)=0.01, P(C)=0.04. Find the probability that at least one of these foul-ups occurs for a randomly chosen order. Hint: P(A U B U C)= P(A) + P(B) + P(C) - P(A ∩ B) – P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C) Example 4. On-Job Training A corporation provides separate classes in reading and practical mathematics for its employees. 40% of workers signed up for the reading classes, and 50% for the practical mathematics classes. Of those signing up for the reading classes, 30% signed up for the mathematics classes. (1) What is the probability that a randomly selected worker signed up for both classes? (2) What is the probability that a randomly selected worker who signed up for the mathematics classes also signed up for the reading classes? (3) What is the probability that a randomly chosen worker signed up for at least one of these two classes? (4) Are the events “Signs up for reading classes” and “Signs up for mathematics classes” statistically independent? Example 5. Loan Default A bank classifies borrowers as high-risk or low-risk. Only 15% of its loans are made to those in the high-risk category. Of all its loans, 5% are in default, and 40% of those in default are to high- risk borrowers. What is the probability that a high-risk borrower will default? A=a loan is made to high-risk category B=a loan is in default P(A)=0.15, P(B)=0.05, P( A | B )=0.4, P(B | A)=?

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conditional probability, conditional probabilities, possible outcomes, sample space, joint probability, marginal probability, probability theory, Multiplication Rule, random variable, random variables

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posted: | 9/17/2010 |

language: | English |

pages: | 11 |

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