University of California Los Angeles Department of Statistic

University of California, Los Angeles Department of Statistics Statistics 10 Instructor: Nicolas Christou Practice exam Problem 1 Three plants, Ci, Ca, and Cs, produce respectively, 10, 50, and 40 percent of a company\'s output. Although plant Ci is a small plant, its manager believes in high quality and only 1 percent of its products are defective. The other two, C2 and Cs produce items that are 3 and 4 percent defective, respectively. All products are sent to a central warehouse. a. An item at the warehouse is selected at random for inspection. What is the probability that this item is defective? b. One item at the warehouse is selected at random and observed to be defective. What is the probability that it came from plant C1? is the probability that it came from plant C,? Problem 2 Part A: Two dice are rolled and the sum of the two numbers shown is observed. a. The two dice are rolled once. You are given the information that the sum is 8. What is the probability that the two numbers were (4,4)? b. The two dice are rolled 13 times. What is the probability of observing at least one sum of 10? c. You roll the two dice 3 times. What is the probability of the following sequence: sum5sum 12 sum-9 Part B: A tennis player A has probability of of winning a set against player B. A match is won by the player who first wins three sets. Find the probability that A wins the match in 4 sets. Problem 3 those from supplier C 90%. A seed from supplier A, 30% from B, and 30% from C and mixes these seeds together. seeds from supplier A have 85% germination rate, those from supplier B 75%, and company purchases 40% of their bean seds packaging a. Find the probability that a randomly selected seed from the mixed seeds will germinate. Denote this probability with P(G). b. Given that a seed germinates, find the probability that the seed was purchased from supplier A Problem 4 Two fair dice are rolled and the sum of the two numbers is observed. What is the probability that a sum of 2 appears before a sum of 6? Problem 5 You are going to play roulette at a casino. As a reminder a roulette has 38 plus 0 and 00). Let\'s say that you want to n times. How large need n be to miake the probability of at least i win at least 99% a reminder a zoulet (1-36 bet on number 13, and you will play this gari game

Solution

1.

Let
D = defective

Thus,

a)

P(D) = P(C1) P(D|C1) + P(C2) P(D|C2) + P(C3) P(D|C3)

= 0.10*0.01 + 0.50*0.03 + 0.040*0.04

= 0.0176 [ANSWER]

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c)

P(C1|D) = P(C1) P(D|C1) / P(D)

= 0.10*0.01/0.0176

= 0.056818182 [ANSWER]

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