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Unformatted text preview: ECE316 Additional Tutorial for the week of June 1-5 Problems from Textbook – 8 th Edition – Chapter 4 – Page 172 Problem 4.1: Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 orange balls. Suppose that we win \$2 for each black ball selected and we lose \$1 for each white ball selected. Let X denote our winnings. What are the possible values of X , and what are the probabilities associated with each value? Solution: • Assume that selecting the orange ball will not affect winning or losing. • Selecting 2 balls out of 8+4+2 = 14 total balls, we will have a total number of combinations of g 14 2 G • The possible values of X are : Balls selected Value of X Number of combinations 2 black 4 g 4 2 G 1 black & 1 orange 2 g 4 1 G × g 2 1 G 1 black & 1 white 1 g 4 1 G × g 8 1 G 2 orange 0 g 2 2 G 1 orange & 1 white -1 g 8 1 G × g 2 1 G 2 white -2 g 8 2 G c The associated probabilities of different values of X ¡¢£ = 4¤ = g 4 2 G g 14 2 G = 6 91 , ¡¢£ = 2¤ = g 4 1 G × g 2 1 G g 14 2 G = 8 91 ¡¢£ = 1¤ = g 4 1 G × g 8 1 G g 14 2 G = 32 91 , ¡¢£ = 0¤ = g 2 2 G g 14 2 G = 1 91 ¡¢£ = −1¤ = g 8 1 G × g 2 1 G g 14 2 G = 16 91 , ¡¢£ = −2¤ = g 8 2 G g 14 2 G = 28 91 Problem 4.2: Two fair dice are rolled. Let X equal the product of the 2 dice, compute gG¡ = ¢£ for ¢ = 1,2, … ,36 Solution: 1 2 3 4 5 6 1 1 2 3 4 5 6 2 2 4 6 8 10 12 3 3 6 9 12 15 18 4 4 8 12 16 20 24 5 5 10 15 20 25 30 6 6 12 18 24 30 36 The corresponding probabilities will be Value of X Number of occurrences Associated probability Value of X Number of occurrences Associated probability 1 1 1 36 ¤ 23 0 36 ¤ 2 2 2 36 ¤ 24 2 2 36 ¤ 3 2 2 36 ¤ 25 1 1 36 ¤ 4 3 3 36 ¤ 26 0 36 ¤ 5 2 2 36 ¤ 27 0 36 ¤ 6 4 4 36 ¤ 28 0 36 ¤ 7 0 36 ¤ 29 0 36 ¤ 8 2 2 36 ¤ 30 2 2 36 ¤ 9 1 1 36 ¤ 31 0 36 ¤ 10 2 2 36 ¤ 32 0 36 ¤ 11 0 36 ¤ 33 0 36 ¤ 12 4 4 36 ¤ 34 0 36...
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