Expected value if two coins are drawn simultaneously from a bag containing $2$ one unit and $3$ two units of money

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This is a question from the Resonance DLPD Algebra book (Page 250). To quote the book:

From a bag containing $2$ one rupee and $3$ two rupees coins, a person is allowed to draw $2$ coins simultaneously; find the value of his expectation.

My try:

There are three cases:

  1. The person draws $2$ one rupee coins.
  2. The person draws one $1$ rupee coin and one $2$ rupees coin.
  3. The person draws two $2$ rupees coin.

The probability of case 1 is:

$$P_1=\frac25*\frac14 = \frac2{20}$$
The value of the two coins is $V_1=2$.

The probability of case 2 is:

$$P_2=\frac25*\frac34 = \frac6{20}$$
The value of the two coins is $V_2=3$.

The probability of case 3 is:

$$P_3=\frac35*\frac24 = \frac6{20}$$
The value of the two coins is $V_3=4$.

Hence his expected value should be:

$$V = P_1V_1+P_2V_2+P_3V_3 = 2.30$$

However, according to the answer given in my book, the correct value is $3.20$! But how? Where did I go wrong with my above approach?

Best Answer

For your case 2, the coins can be drawn in either order, so you need to multiply by 2.

Another approach:

The expected value of one coin is $$\frac{2 \times 1 + 3 \times 2}{5} = \frac{8}{5}$$

so the expected value of two coins is $2 \times (8/5) = 16/5$.

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