Unformatted text preview: Department of Mathematics Name: ________________________________ Poly ID: _______________________________ Instructor's Name: _______________________ Course No:_____________________________ Course Section: _________________________ Homework Section No: ___________________ Willis Thai 1.11 MA2212 HW01 Describe the outcome space for each of the following experiments: (a) A student is selected at random from a statistics class, and the student's ACT score in mathematics is determined. HINT: ACT test scores in mathematics are integers between 1 and 36, inclusive. S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36} (b) A candy bar with a 20.4gram label weight is selected at random from a production line and is weighed. S = {20.4} (c) A coin is tossed three times, and the sequence of heads and tails is observed. S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} Describe the outcome space for each of the following experiments: (a) Consider families that have three children each and select one such family at random. Describe the sample space S in terms of their children as 3tuples, agreeing for example, "gbb" would indicate that the youngest is a girl and the two oldest are boys. S = {ggg, ggb, gbg, gbb, bgg, bgb, bbg, bbb} (b) A rat is selected at random from a cage, and its sex is determined. S = {M, F} (c) A state selects a threedigit integer at random for one of its lottery games. S = {100, 101, 102, 103, ..., 999} A field of beans is planted with three seeds per hill. For each hill of beans, let Ai be the event that I seeds germinate, I = 0, 1, 2, 3. Suppose that P(A0) = 1/64, P(A1) = 9/64, and P(A2) = 27/64. Give the value of P(A3). P(A0) + P(A1) + P(A2) + P(A3) = 1 1  P(A0)  P(A1)  P(A2) = P(A3) 1 1/64 9/64 27/64 = 27/64 A fair eightsided die is rolled once. Let A = {2, 4, 6, 8}, B = {3, 6}, C = {2, 5, 7}, and D = {1, 3, 5, 7}. Assume that each face has the same probability. (a) Give the values of: (i) P(A) = .5 (ii) P(B) = .25 (iii) P(C) = .375 (iv) P(D) = .5 (b) Give the values of: (i) W{V V{ = .125 (ii) W{V V{ = 0 (iii) W{V { = .25 (c) Using, Theorem 1.25, give the values of: { { { {. { (i) W{V V{ { = .5 + .25  .125 = .625 1.12 1.23 1.25 Willis Thai (ii) W{V V{ { (iii) W{V 1.29 MA2212 { { { {. { { { { {. { { = .25 + .375 0 = .625 { = .375 + .5  .25 = .625 HW01 Roll a fair sixsided die three times. Let A1 = {1 or 2 on the first roll}, A2 = {3 or 4 on the second roll}, and A3 = {5 or 6 on the third roll}. It is given that P(Ai) = 1/3, I = 1, 2, 3; W{VX V { = V V { = (1/3)3. (1/3)2, I j; and W{V (a) Use Theorem 1.26 to find W{V V V {. { # { #{  { ${  { %{ . { # $ %{ ${ . { # %{ . { $ %{ {
# $ %{ # %   . %%. . . . .
$ # # # % # % # $ % # $ % . .
$ # $ % # $ % . .
$ # $ % # $ %  # % % # % % # $ (b) Show that W{V { {
# # # # # $ $ $ $ $ $ %{ %{ %{ %{ %{ %{ V # % V { = 1 (1 1/3)3. F $ F F
% F . F % { { { F . . . F . F $ { # Q.E.D. F . $ F % % F = 1 (1 1/3)3 1.212 Let x equal a number that is selected randomly from the closed interval from zero to one, [0, 1]. Use your intuition to assign values to: (b) W{{Y (a) W{{Y 3 Y 3 {{. {{. 3 Y 3 {{. {{.
# $ # % $ % (c) W{{Y Y (d) W{{Y V V 1.217 Let W (a) If W{V { { #{  { ${   { { { { { {  { { { { { { V V W{V { V , where events A1, A2, ..., Am are mutually exclusive and exhaustive. W{V {, show that W{VX { , I = 1, 2, ..., m. (b) If V VX, where h < m, and (a) holds, prove that P(A) = h/m. Willis Thai {
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 Fall '08
 Yang
 Math, Probability theory, Christopher Nolan, Randomness, Dice

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