Problem 35 - A First Course in Probability

Question

A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the same color, then you win \(\$ 1.10 ;\) if they are different colors, then you win \(-\$ 1.00 .\) (That is, you lose \(\$ 1.00 .\) ) Calculate (a) the expected value of the amount you win; (b) the variance of the amount you win.

Step-by-Step Solution

Answer

The expected value of the amount you win is $0.10, and the variance of the amount you win is approximately $0.892.

Step 1: Calculate the probability of drawing the same color marbles

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) = \( \frac{2}{9} + \frac{2}{9} = \frac{4}{9} \)

Step 2: Calculate the probability of drawing different color marbles

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 - \( \frac{4}{9} = \frac{5}{9} \)

Step 3: Calculate the expected value of the amount you win

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

Step 4: Calculate the variance of the amount you win

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.