Respuesta :
Answer:
So, the probability that you rolled the 8-sided die given that you rolled a 1 is approximately 0.296, or 29.6%.
Answer:
To find the probability of rolling the 8-sided die given that you rolled a 1, we can use Bayes' theorem, which states:
\[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \]
Where:
- \( P(A|B) \) is the probability of event A occurring given that event B has occurred.
- \( P(B|A) \) is the probability of event B occurring given that event A has occurred.
- \( P(A) \) is the probability of event A occurring.
- \( P(B) \) is the probability of event B occurring.
In this scenario:
- Event A is rolling the 8-sided die.
- Event B is rolling a 1.
We know that the probability of rolling a 1 on the 8-sided die is \( \frac{1}{8} \).
We also know that the probability of rolling a 1 on the 11-sided die and the 15-sided die is \( \frac{1}{11} \) and \( \frac{1}{15} \) respectively.
Given that you rolled a 1, the total probability of rolling a 1 is:
\[ P(B) = P(B|A) \cdot P(A) + P(B|A') \cdot P(A') \]
Where:
- \( P(B|A) \) is the probability of rolling a 1 given that you rolled the 8-sided die, which is \( \frac{1}{8} \).
- \( P(A) \) is the probability of rolling the 8-sided die, which is \( \frac{1}{3} \) (since there are three dice).
- \( P(B|A') \) is the probability of rolling a 1 given that you didn't roll the 8-sided die, which is \( \frac{1}{11} + \frac{1}{15} \).
- \( P(A') \) is the probability of not rolling the 8-sided die, which is \( \frac{2}{3} \).
\[ P(B) = \frac{1}{8} \cdot \frac{1}{3} + \left( \frac{1}{11} + \frac{1}{15} \right) \cdot \frac{2}{3} \]
\[ P(B) = \frac{1}{24} + \left( \frac{1}{11} + \frac{1}{15} \right) \cdot \frac{2}{3} \]
Now, we can calculate \( P(A|B) \) using Bayes' theorem:
\[ P(A|B) = \frac{\frac{1}{8} \cdot \frac{1}{3}}{\frac{1}{24} + \left( \frac{1}{11} + \frac{1}{15} \right) \cdot \frac{2}{3}} \]
\[ P(A|B) = \frac{\frac{1}{24}}{\frac{1}{24} + \left( \frac{1}{11} + \frac{1}{15} \right) \cdot \frac{2}{3}} \]
\[ P(A|B) = \frac{\frac{1}{24}}{\frac{1}{24} + \left( \frac{13}{165} \right) \cdot \frac{2}{3}} \]
\[ P(A|B) = \frac{\frac{1}{24}}{\frac{1}{24} + \frac{13}{82}} \]
\[ P(A|B) = \frac{\frac{1}{24}}{\frac{35}{264}} \]
\[ P(A|B) = \frac{264}{840} \]
\[ P(A|B) = \frac{22}{70} \]
\[ P(A|B) = \frac{11}{35} \]
So, the probability that you rolled the 8-sided die given that you rolled a 1 is \( \frac{11}{35} \).
