STAT_333_midterm_1 - STAT 333 Test 1 SOLUTION October 5,...

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STAT 333 Test 1 SOLUTION October 5, 2010 1. [5] In a small factory, there are 4 machines which package the same product. The per- centages that these machines package are 15%, 20%, 30% and 35%. Also the probabilities that these machines can damage the product during packaging are 0 . 05, 0 . 04, 0 . 03, and 0 . 02, respectively. All the packages are subsequently mixed up. (a)[2] If we randomly pick out one package, find the probability that the product it contains is damaged. (b)[3] If the product in the package selected is damaged, find the probability that this product was packaged by machine 4. Solution : (a) Define A = { the selected package contained a damaged product } , B i = { the product was packaged by machine i } , i = 1 , 2 , 3 , 4. Then by the total probability rule, we have P ( A ) = 4 X i =1 P ( A | B i ) P ( B i ) = 0 . 15 × 0 . 05 + 0 . 2 × 0 . 04 + 0 . 3 × 0 . 03 + 0 . 35 × 0 . 02 = 0 . 0325 (b)By the Bayes’ formula P ( B 4 | A ) = P ( A | B 4 ) P ( B 4 ) 4 i =1 P ( A | B i ) P ( B i ) = 0 . 02 × 0 . 35 0 . 0325 = 0 . 215 2. [15] Suppose the number of customers coming to a gas station follows a Poisson distribution at a rate of 5 per hour. (a) [5] What is the probability of having no more than 1 customer in a 30-minute period? (b) [5] If there are no customers coming to the gas station between 2:30pm and 3:30pm, find the probability that no customers come to the gas station between 2:30pm and 4:30pm. ( Hint:
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This note was uploaded on 01/19/2011 for the course STATISTICS STAT 333 taught by Professor Menzhongxian during the Fall '10 term at Waterloo.

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STAT_333_midterm_1 - STAT 333 Test 1 SOLUTION October 5,...

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