91Ó°ÊÓ

To calculate the probability of drawing 2 marbles of the same color, we consider two cases: drawing 2 red marbles and drawing 2 blue marbles. Let's denote these probabilities as P(RR) and P(BB), respectively. For drawing 2 red marbles: P(RR) = probability of drawing the first red marble * probability of drawing the second red marble, given that the first red marble is already drawn \( P(RR) = \frac{5}{10} \times \frac{4}{9} = \frac{2}{9} \) Similarly, for drawing 2 blue marbles: P(BB) = probability of drawing the first blue marble * probability of drawing the second blue marble, given that the first blue marble is already drawn \( P(BB) = \frac{5}{10} \times \frac{4}{9} = \frac{2}{9} \) Now, the probability of drawing marbles of the same color is the sum of the probabilities of drawing 2 red marbles and 2 blue marbles. P(same color) = P(RR) + P(BB) = \( \frac{2}{9} + \frac{2}{9} = \frac{4}{9} \)

To find the probability of drawing marbles of different colors, we can subtract the probability of drawing marbles of the same color from 1 (since the total probability of all events must be 1) P(different color) = 1 - P(same color) = 1 - \( \frac{4}{9} = \frac{5}{9} \)

The expected value is the sum of the product of each event's probability and the corresponding amount won. Let's denote the expected value as E(X). E(X) = P(same color) * \(1.10 + P(different color) * (-\)1.00) E(X) = \(\frac{4}{9}\) * \(1.10 - \(\frac{5}{9}\) * \)1.00 E(X) = $ 0.10

To calculate the variance, we need to find the squared deviation of each event's amount won from the expected value and multiply it by the corresponding event's probability. Let's denote the variance as Var(X). Var(X)= P(same color) * (1.10 - E(X))^2 + P(different color) * (-1.00 - E(X))^2 Var(X) = \(\frac{4}{9}\) * (1.10 - 0.10)^2 + \(\frac{5}{9}\) * (-1.00 - 0.10)^2 Var(X) = \(\frac{4}{9}\) * (1.00)^2 + \(\frac{5}{9}\) * (-1.10)^2 Var(X) = \(\frac{4}{9}\) * 1 + \(\frac{5}{9}\) * 1.21 Var(X) ≈ $ 0.892 So, (a) the expected value of the amount you win is \(0.10, and (b) the variance of the amount you win is approximately \)0.892.