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A card is drawn from a deck of 52. The score equal to its rank unless it is a court card (Jack, Queen or King) with a score of 10, otherwise equal to its rank and Ace counts as one.

What is the expected value of the score?

I am new to this and a similar example on how to solve this would be great.

user133110
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2 Answers2

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Assuming that a court card has a value of $10$ then the expected value of the score of one card is simply the average of the scores of one suit's cards:

$$E(X) = \frac{1+2+3+4+5+6+7+8+9+10+10+10+10}{13} = 85/13.$$

John
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The expected value is just the average over the full deck (if you assume each card equally likely). In this case, it is just the average value of each suit, i.e.: $$ \frac{1}{13} \left(\sum_{1 \le k \le 10} k + 3 \cdot 10 \right) = \frac{10 (10 + 1)}{2 \cdot 13} + \frac{3 \cdot 10}{13} = \frac{85}{13} $$

vonbrand
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    You lost a factor $2$ in the denominator if you are assuming the court cards count zero. OP just edited to show they count for $10$ – Ross Millikan Mar 04 '14 at 21:40