Theory of Expectation :: Drawing Balls, Cards, Items, Products

Solution

"x" indicate the amount got by the man [Since we are required to find the mans expectation, the variable would represent the amount got by the man] The amount got by the man would be Rs. 2 if he draws "0 White and 2 Black" balls Rs. 3 if he draws "1 White and 1 Black" ball Rs. 4 if he draws "2 White and 0 Black" balls ⇒ The values carried by the variable ("x") would be either 2, 3 or 4 ⇒ "X" is a discrete random variable with range = {2, 3, 4} "X" represents the random variable and P(X = x) represents the probability that the value within the range of the random variable is a specified value of "x" Total number of balls = 5 White + 7 Black = 12 Total Number of possible choices (in drawing the two balls) = Number of ways in which the two balls can be drawn from the total 12 = 12Cs ⇒ n = 12 × 11 2 × 1 = 66 Probabilty that the two balls drawn would be "0 White and 2 Black" ⇒ P(0W2B) = 5C0 × 7C2 12C2 White Black Total Available 5 7 12 To Choose 0 2 2 Choices 5C0 7C2 12C2 = 1 × 7 × 6 2 × 1 66 = 21 66 = 7 22 "1 White and 1 Black" ⇒ P(1W1B) = 5C1 × 7C1 12C2 White Black Total Available 5 7 12 To Choose 1 1 2 Choices 5C1 7C1 12C2 = 5 1 × 7 1 66 = 35 66 "2 White and 0 Black" ⇒ P(2W0B) = 5C2 × 7C0 12C2 White Black Total Available 5 7 12 To Choose 2 0 2 Choices 5C2 7C0 12C2 = 5 × 4 2 × 1 × 1 66 = 10 66 Probability for the amount got by the person to be Rs. 2 ⇒ P(X = 2) = P(0W2B) = 21 66 Rs. 3 ⇒ P(X = 3) = P(1W1B) = 35 66 Rs. 4 ⇒ P(X = 4) = P(2W0B) = 10 66 The probabilty distribution of "x" would be x 2 3 4 P(X = x) 21 66 35 66 10 66 Calculations for Mean and Standard Deviations x P (X = x) px [x × P (X = x)] x2 px2 [x2 × P (X = x)] 2 21 66 42 66 4 84 66 3 35 66 105 66 9 315 66 4 10 66 40 66 16 160 66 Total 1 187 66 559 66 = 2.83 = 8.47 The mans expectation ⇒ Expectation of "x" ⇒ E (x) = Σ px = Rs. 2.83 Variance of the amount got by the man ⇒ var (x) = E (x2) − (E(x))2 ⇒ var (x) = Σ px2 − (Σ px)2 = 8.47 − (2.83)2 = 8.47 − 8.0083 = 0.4617 Standard Deviation of the amount got by the man ⇒ SD (x) = + √ Var (x) = + √ 0.4617 = + 0.679