Two standard dice are rolled. What is the expected number of 1's obtained? Express your answer as a common fraction.
The probability of rolling a number other than 1 on one die is $\frac{5}{6}$, so the probability of rolling zero 1's on two dice is $\left(\frac{5}{6}\right)^2 = \frac{25}{36}$. The probability of rolling a 1 on one die is $\frac{1}{6}$, so the probability of rolling 1's on two dice (for a total of two 1's) is $\left(\frac{1}{6}\right)^2 = \frac{1}{36}$. Since the only other possibility is rolling exactly one 1, the probability of rolling one 1 is $1-\frac{25}{36}-\frac{1}{36} = \frac{10}{36}$. We find the expected number of 1's to be $E = \frac{1}{36} \cdot 2 + \frac{10}{36} \cdot 1 + \frac{25}{36} \cdot 0 = \boxed{\frac{1}{3}}$.
\frac{1}{3}
Level 4
Counting & Probability
Problem: Two standard dice are rolled. What is the expected number of 1's obtained? Express your answer as a common fraction.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the expected number of 1's when rolling two standard dice, we first define the random variable X as the number of 1's rolled. The possible values for X are 0, 1, or 2. 

Next, we calculate the probability of each outcome. The probability of rolling a 1 on a single die is 1/6, while the probability of not rolling a 1 is 5/6. 

For X = 0 (no 1's), the probability is:
P(X = 0) = (5/6) * (5/6) = 25/36.

For X = 1 (one 1), the probability is:
P(X = 1) = (1/6) * (5/6) + (5/6) * (1/6) = 2 * (1/6) * (5/6) = 10/36 = 5/18.

For X = 2 (two 1's), the probability is:
P(X = 2) = (1/6) * (1/6) = 1/36.

Now we have the probabilities:
P(X = 0) = 25/36,
P(X = 1) = 5/18,
P(X = 2) = 1/36.

Next, we calculate the expected value E(X) using the formula:
E(X) = Σ [x * P(X = x)] for all x.

Calculating each term:
E(X) = 0 * (25/36) + 1 * (5/18) + 2 * (1/36).

This simplifies to:
E(X) = 0 + (5/18) + (2/36).

To combine the fractions, we convert (2/36) to (1/18):
E(X) = (5/18) + (1/18) = (6/18) = 1/3.

Thus, the expected number of 1's obtained when rolling two standard dice is 1/3.<|END_OF_LATENT|> The final answer is $\frac{1}{3}$.