Certainly! Let's tackle each part of the question step-by-step.
We have a 12-sided solid (or die) with faces numbered from 1 to 12. Since all faces are equal in size, each number has an equal probability of being rolled.
### Part (a): Find the Probability of Rolling a Number Greater than 10
1. Identify Favorable Outcomes:
Numbers greater than 10 on a 12-sided die are 11 and 12. So there are 2 favorable outcomes.
2. Total Possible Outcomes:
There are 12 faces on the die, so there are 12 possible outcomes.
3. Calculate Probability:
The probability [tex]\( P \)[/tex] of rolling a number greater than 10 is given by the ratio of favorable outcomes to the total outcomes:
[tex]\[
P(\text{number greater than 10}) = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{2}{12}
\][/tex]
4. Simplify the Fraction:
[tex]\[
\frac{2}{12} = \frac{1}{6} \approx 0.1667
\][/tex]
So, the probability of rolling a number greater than 10 is approximately 0.1667 (or [tex]\( \frac{1}{6} \)[/tex]).
### Part (b): Find the Probability of Rolling a Number Less than 5
1. Identify Favorable Outcomes:
Numbers less than 5 on a 12-sided die are 1, 2, 3, and 4. So there are 4 favorable outcomes.
2. Total Possible Outcomes:
Again, there are 12 faces on the die, so there are 12 possible outcomes.
3. Calculate Probability:
The probability [tex]\( P \)[/tex] of rolling a number less than 5 is given by the ratio of favorable outcomes to the total outcomes:
[tex]\[
P(\text{number less than 5}) = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{4}{12}
\][/tex]
4. Simplify the Fraction:
[tex]\[
\frac{4}{12} = \frac{1}{3} \approx 0.3333
\][/tex]
So, the probability of rolling a number less than 5 is approximately 0.3333 (or [tex]\( \frac{1}{3} \)[/tex]).
### Notes on Expectation:
The phrase "expect either a 4, 6, or 9 to be rolled" appears ambiguous in the question context unless further qualified. If the question is part of the original prompt, then it might be asking for expectations in a different scenario, unrelated to probability calculations directly here. Rolling either a 4, 6, or 9 would have a different set of considerations.
To summarize:
- The probability of rolling a number greater than 10 is 0.1667.
- The probability of rolling a number less than 5 is 0.3333.
