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 standard deviation of the score?
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Let $X$ be the score. The standard deviation of the score is the square root of the variance. And the variance is $E(X^2)-(E(X))^2$.
We assume that you know how to calculate $E(X)$. For $E(X^2)$, note that $X^2=1^2,2^2,3^2,\dots, 9^2$ each with probability $\frac{1}{13}$, and $10^2$ with probability $\frac{4}{13}$. Thus $$E(X^2)=\frac{1}{13}\left(1^2+2^2+\cdots+9^2 +4\cdot 10^2\right).$$
André Nicolas
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