Tutorial Workshop 2 Solution

Tutorial Workshop 2 Solution - ECON 1203 Tutorial Sample...

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1 ECON 1203 Tutorial Sample Solutions Semester 2 2015 Weeks 3 and 4 1. (a) Explain what it means to say that two probabilistic events in a sample space are mutually exclusive of one another. If two events – let’s call them A and B – are mutually exclusive, then it means that they do not have any simple events in common: i.e., that the simple events that combine to make up A have no elements in common with those that make up B. (a) Explain what it means to say that two probabilistic events in a sample space are independent of one another. When two events are independent of one another, it means that the effect of conditioning on the occurrence of one of them has no effect of the marginal probability of the other: i.e., Pr(A / B) = Pr(A). (b) Why can two events not at the same time be both mutually exclusive and independent of one another? Because if A and B are mutually exclusive, then Pr(A and B) = 0, whereas if they are independent, Pr(A and B) = Pr(A)*Pr(B) 0.
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2 2. A department store wants to study the relationship between the way customers pay for an item and the price of the item. 250 transactions are recorded and the following table is formed. Price category Payment Cash Credit card Debit card Under $20 $20-$100 Over $100 15 11 6 9 53 38 18 52 48 Convert the table to a joint distribution. Express each of the following questions in terms of probability statements, and then solve: Joint distribution: Price category Means of Payment Cash Credit card Debit card Marginal Under $20 0.06 0.036 0.072 0.168 $20-$100 0.044 0.212 0.208 0.464 Over $100 0.024 0.152 0.192 0.368 Marginal 0.128 0.4 0.472 1 (a) What is the probability that an item is under $20? P(Under $20) = 0.168 (b) What is the probability that an item with a price tag of $43 is paid for in cash? P($20-$100 and cash) = 0.044 (c) What is the probability that people pay for an item that is at least $20 by credit? P( $20 and credit) = 0.212 + 0.152 = 0.364 (d) If somebody used a debit card to pay for an item, what is the probability that the item was less than $100? P(<$100|debit) = (0.072+0.208)/0.472 = 0.593 (e) Are price and means of payment independent? One way to check is to compare the marginal distribution of price with the conditional distribution of price given a particular payment type (say, cash): P($20-$100|cash) = 0.344 P(($20-$100) = 0.464 This implies dependence. 3. In a small batch of 20 manufactured widgets, there are, in fact, 3 defective ones. You, as quality control officer for the company making the widgets, decide to examine a sample of 3 widgets, selected without replacement, to see how many defective ones are selected. (a) Use a probability tree to evaluate the probability distribution of the number of defectives sampled.
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3 The tree is of the obvious kind with the first branch from a branch where the probability of defective is 0.15 and not defective is 0.85. From the upper of these branches at the next node the probability of defective being selected is 2/19 and non-defective is 17/19. From the lower first branch, the probability of a defective is 3/19 and of a non-defective is 16/19. From the nodes at
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  • Three '14
  • Probability, Probability distribution, Probability theory, Sharpe

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