I've been stuck on this question for a while. Two coins are selected at random from $3$ pennies, $2$ nickels, and $1$ dime with no replacement, and $X$ is the sum of the two coins. What is the probability function and the expected value?
My solution: For probability function for $X$,
$$f(x)=\begin{cases}
\frac36\cdot\frac25,&\text{for }X=2\text{ cents}\\\\
\frac26\cdot\frac25,&\text{for }X=6\text{ cents}\\\\
\frac36\cdot\frac15,&\text{for }X=11\text{ cents}\\\\
\frac36\cdot\frac15,&\text{for }X=10\text{ cents}\\\\
\frac16\cdot\frac35,&\text{for }X=15\text{ cents}\;.
\end{cases}$$
I can not seem to get a reasonable solution for $\mathrm{E}X$, because $\mathrm{E}X$ is a mean value where each value that $X$ takes on is either $2, 6, 10, 11$, or $15$ cents. I can't spend any more time on this, it's driving me nuts. Thanks!
Best Answer
You are thinking in the right direction, but if you check, your probabilities don't add to $1$. For 11 cents, $\frac 36 \cdot \frac 15$ is the chance you take a penny and then a dime, but you can also take the dime first, so it needs to be doubled. For 15, you need a dime and a nickel, but in either order, so $2\cdot \frac 16 \cdot \frac 25$. 10 cents is wrong differently. Then EX is just $\sum P(X)X$, so $2P(2)+6P(6)+\ldots$