Brent rolls a fair dodecahedral die with numbers $1,2,3,...,12$ on its faces. What is the expected number of digits in the number he obtains? Express your answer in a decimal.
I need to find the expected value of a random variable that depends on the outcome of the die roll. The random variable is the number of digits in the number rolled. To find the expected value, I need to multiply each possible value of the random variable by its probability, and then add them up. The possible values of the random variable are $1$ and $2$, since the numbers on the die have either one or two digits. The probability of getting a one-digit number is $\frac{9}{12}$, since there are $9$ one-digit numbers on the die: $1,2,3,...,9$. The probability of getting a two-digit number is $\frac{3}{12}$, since there are $3$ two-digit numbers on the die: $10,11,12$. So the expected value is $1 \cdot \frac{9}{12} + 2 \cdot \frac{3}{12}$. Simplifying, I get $\frac{9}{12} + \frac{6}{12} = \frac{15}{12} = 1.25$.

