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!"#$%!"""""""""""""""""""""""""""""""""""""""""! ! Probability and Statistics Test 1 Solutions ! 1. The probability that student A will fail a certain statistics examination is 0.5, the probability that student B will fail the examination is 0.2, and the probability that both student A and student B will fail the examination is 0.1. a. What is the probability that at least one of these two students will fail the examination? P(A) + P(B) - P(A!B) = 0.5 + 0.2 - 0.1 = 0.6 b. What is the probability that neither student A nor student B will fail the examination? 1 - ( P(A) + P(B) - P(A!B) )= 1 – 0.6 = 0.4 c. What is the probability that exactly one of the two students will fail the examination? P(A) + P(B) – 2(P(A!B)) = 0.5 + 0.2 – 2X0.1 = 0.5 2. In how many different ways can the five letters a, b, c, d, and e be arranged? 5! 3. A school contains students in grades 1, 2, 3, 4, 5, and 6. Grades 2, 3, 4, 5, and 6 all contain the same number of students, but there is twice this number in grade 1. a. If a student is selected at random from a list of all the students in the school, what is the probability that she will be in grade 3? Total number of students = 7x Number in grade 3 = x P(grade 3) = x/7x = 1/7 b. What is the probability that the selected student will be in an oddnumbered grade? Total number of students = 7x Number in grades 1, 3, and 5 = 4x P(odd-numbered grade) = 4x/7x = 4/7 4. An elevator in a building starts with five passengers and stops at seven floors. If every passenger is equally likely to get off at each floor and all the passengers leave independently of each other, what is the probability that no two passengers will get off at the same floor? ! !"#$%!"""""""""""""""""""""""""""""""""""""""""! ! ! !