Question

2. An urn contains six white balls and four black balls. Two balls are randomly selected from the urn. Let X represent the number of black balls selected. (a) Identify the probability distribution of X. State the values of the parameters corresponding to this distribution (b) Compute P(X = 0), P(X= 1), and P(X= 2). (c) Consider a game of chance where you randomly select two balls from the urn. You then win $2 for every black ball selected and lose $1 for every white ball selected. Let Y represent the total amount won. Represent the probability distribution of Y as a probability table. (d) Find the mean and standard deviation of Y. Would you play this game? Explain.

Answer 1

Griven oi 2-0 things from (3) = 1! 61 0 10 balls. Way to select 2 balls from solution win contains white balls = 6 @ black balls - 4 Total = 10 Two balls are randomly selected from the win. => Let x represent the number of black balls selected. i x has possible values 0,1,2. »P6749 means both balls are white. We use concept of Combination to find probability. do PCX =0) = Number of way. to select 2 balls Combination ;s Number of - white from 6 white balls. way to select a Number of way to select 2 balls n things.is, from 10 balls. 2! (1-2)] (2) (10) 2! (6-2)! 21x4! 뭥 80 90 = 0.33 Number of way to gelect 1 white Number of way to select 1. balls from 6 white balls. black balls from 4 black balls. 4 X 6x4 6x4 10! 2! (10-2)) 6*4x1x2x8! 0.54 10x9x85 Gé X 101 2!(10-2) 21X8! 101 6x5x41 X 10x9%81 or X P(X=1) X 21x8! 11 101 (9)

As 2}(4-2)! 11 28x28 PCX=2) = Number of way to select 2 balls from 4 bläck balls. Number of way to select 2 balls from 10 balls. (1) 9.4 2*8! (2) 10! 101 2! (10-2)! 4x3X2%. 2 рухи 4 x3 10x9 ởo P(X=0) = 0.33, PCX-1) = 0.54, P(x - 2) = 0.13 $2 X 0 (b) = 0.13 © for -) we make Y (minus sign indicate loss). -$4. Selecting black ball, you win fost selecting white ball, you loses $1 fost every white ball. 8091 every black ball. Let y represent the total amount won? below table 40e amount of won. X 0 (2 white balls, o black balls) 2x-1) + 0x2 -$-2 111 white balls, 1 black balls) 1x6-1) + 1x2 = $ 1. 2 Co white balls, 2 black balls) Ox61) + 2x2 oo probability distribution of y is below . P(x=0) p(y=-2) = 0.33 p(X=1) = P(Y= 1) P(x=2) = P(y=4) formula 1091 mean : + Mean = Eyi plyi) -(-2)(0.33) +(1)(0.54) +C4) (012) 0.4.Cyes, we play this game because Standard deviation Eye Plyi) – (Mean) mean is ° positive) (1810.38) + (120.54)+(4)*0.13) - (0.434 0.54 0.13 @ v1.14 = 1.068 ::Standased deviation 11.066 (1.3) -0.16 -X

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