To determine the probability of rolling a sum of either 7 or 11 with two six-sided dice, we will follow these steps:
1. Identify the given probabilities:
- The probability of rolling a sum of 7, given as [tex]\( P(7) = \frac{1}{6} \)[/tex].
- The probability of rolling a sum of 11, given as [tex]\( P(11) = \frac{1}{18} \)[/tex].
2. Calculate the total probability of rolling a sum of 7 or 11:
Since the two events (rolling a 7 and rolling an 11) are mutually exclusive (they cannot occur simultaneously because you can only get one sum per roll), we can add these probabilities together:
[tex]\[
P(\text{Sum is 7 or 11}) = P(7) + P(11)
\][/tex]
3. Substitute the given probabilities into the equation:
- [tex]\( P(7) = \frac{1}{6} \approx 0.16666666666666666 \)[/tex]
- [tex]\( P(11) = \frac{1}{18} \approx 0.05555555555555555 \)[/tex]
So,
[tex]\[
P(\text{Sum is 7 or 11}) = 0.16666666666666666 + 0.05555555555555555 = 0.2222222222222222
\][/tex]
4. Summarize the findings:
- The probability of rolling a sum of 7 is [tex]\( \approx 0.16666666666666666 \)[/tex].
- The probability of rolling a sum of 11 is [tex]\( \approx 0.05555555555555555 \)[/tex].
- Therefore, the total probability of rolling a sum of 7 or 11 is [tex]\( \approx 0.2222222222222222 \)[/tex].
Thus, the probability that the sum of the numbers on two rolled dice is either 7 or 11 is approximately [tex]\( 0.2222222222222222 \)[/tex].
