Two black chips and three red chips are put into a bag. two points are awarded for each black chip drawn, and one point is lost for each red chip drawn. what is the expected value for each round if there are two draws per round and the chips are replaced after each draw?

The answer is 0.4.

The values are:
- for the black chip : x₁ = 2
- for the red chip: x₂ = -1

Let's first calculate the possibilities of each chip. There are in total 5 chips (two black chips and three red chips) in the bag
 - the possibility to draw the black chip is 2 out of 5:    P₁ = 2/5
 - the possibility to draw the red chip is 3 out of 5:       P₂ = 3/5

In this example we have 4 different events:
1. Drawing of two black chips: P₃ = P₁ · P₁ = 2/5 · 2/5 = 4/25
2. Drawing of one black chip and then one red chip: P₄ = P₁ · P₂ = 2/5 · 3/5 = 6/25
3. Drawing of one red chip and then one black chip: P₅ = P₂ · P₁ = 3/5 · 2/5 = 6/25
5. Drawing of two black chips: P₆ = P₂ · P₂ = 3/5 · 3/5 = 9/25


Therefore, the expected value for each round if there are two draws per round and the chips are replaced after each draw is 0.4:
P = (x₁ + x₁) · P₃ + (x₁ + x₂) · P₄ + (x₁ + x₂) · P₅ + (x₂ + x₂) · P₆
P = (2+2) · 4/25 + (2-1) · 6/25 + (2-1) · 6/25 + (-1 + -1) · 9/25
P = 4 · 4/25 + 1 · 6/25 + 1 · 6/25 + -2 · 9/25
P = 16/25 + 6/25 + 6/25 - 18/25
P = 28/25 - 18/25
P = 10/25 = 0.4

No solution : there is no number of boxes that she bought

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