Find the best answers to your questions with the help of IDNLearn.com's knowledgeable users. Discover the information you need from our experienced professionals who provide accurate and reliable answers to all your questions.
Sagot :
To find the probability that the coin showed a head given that Jamie scores 6, we can break this problem down using Bayes' theorem. Let's go through the solution step by step.
1. Define Events:
- Let [tex]\( H \)[/tex] be the event that the coin shows a head.
- Let [tex]\( T \)[/tex] be the event that the coin shows a tail.
- Let [tex]\( S_6 \)[/tex] be the event that the score is 6.
2. Given Probabilities:
- The probability that the coin shows a head: [tex]\( P(H) = \frac{1}{3} \)[/tex].
- The probability that the coin shows a tail: [tex]\( P(T) = 1 - P(H) = \frac{2}{3} \)[/tex].
3. Dice Outcomes:
- The dice faces are numbered [tex]\([1, 1, 2, 3, 3, 4]\)[/tex].
4. Calculate Scores:
- If the coin shows a head (event [tex]\( H \)[/tex]), the score is the sum of the two dice.
- If the coin shows a tail (event [tex]\( T \)[/tex]), the score is the product of the two dice.
5. Count Occurrences for Head and Tail:
- Calculate the number of ways to achieve a score of 6 when the coin shows a head and when it shows a tail.
6. Probability of Scoring 6 Given Head:
- There are 6 occurrences where the sum of two dice is 6.
- There are [tex]\( 6 \times 6 = 36 \)[/tex] total possible outcomes for two dice.
- Thus, [tex]\( P(S_6 | H) = \frac{6}{36} = \frac{1}{6} \)[/tex].
7. Probability of Scoring 6 Given Tail:
- There are 4 occurrences where the product of two dice is 6.
- There are [tex]\( 6 \times 6 = 36 \)[/tex] total possible outcomes for two dice.
- Thus, [tex]\( P(S_6 | T) = \frac{4}{36} = \frac{1}{9} \)[/tex].
8. Total Probability of Scoring 6:
- Use the law of total probability:
[tex]\[ P(S_6) = P(S_6 | H) \cdot P(H) + P(S_6 | T) \cdot P(T) \][/tex]
- Substituting the known values:
[tex]\[ P(S_6) = \left( \frac{1}{6} \cdot \frac{1}{3} \right) + \left( \frac{1}{9} \cdot \frac{2}{3} \right) = \frac{1}{18} + \frac{2}{27} = \frac{1}{18} + \frac{2}{27} = \frac{3}{54} + \frac{4}{54} = \frac{7}{54} \approx 0.12963 \][/tex]
9. Bayes' Theorem:
- To find [tex]\( P(H | S_6) \)[/tex], we use Bayes' theorem:
[tex]\[ P(H | S_6) = \frac{P(S_6 | H) \cdot P(H)}{P(S_6)} \][/tex]
- Substituting the values:
[tex]\[ P(H | S_6) = \frac{\left( \frac{1}{6} \right) \cdot \left( \frac{1}{3} \right)}{\frac{7}{54}} = \frac{\frac{1}{18}}{\frac{7}{54}} = \frac{1}{18} \cdot \frac{54}{7} = \frac{3}{7} \approx 0.42857 \][/tex]
Therefore, the probability that the coin showed a head given that Jamie scores 6 is approximately [tex]\(\boxed{0.42857}\)[/tex].
1. Define Events:
- Let [tex]\( H \)[/tex] be the event that the coin shows a head.
- Let [tex]\( T \)[/tex] be the event that the coin shows a tail.
- Let [tex]\( S_6 \)[/tex] be the event that the score is 6.
2. Given Probabilities:
- The probability that the coin shows a head: [tex]\( P(H) = \frac{1}{3} \)[/tex].
- The probability that the coin shows a tail: [tex]\( P(T) = 1 - P(H) = \frac{2}{3} \)[/tex].
3. Dice Outcomes:
- The dice faces are numbered [tex]\([1, 1, 2, 3, 3, 4]\)[/tex].
4. Calculate Scores:
- If the coin shows a head (event [tex]\( H \)[/tex]), the score is the sum of the two dice.
- If the coin shows a tail (event [tex]\( T \)[/tex]), the score is the product of the two dice.
5. Count Occurrences for Head and Tail:
- Calculate the number of ways to achieve a score of 6 when the coin shows a head and when it shows a tail.
6. Probability of Scoring 6 Given Head:
- There are 6 occurrences where the sum of two dice is 6.
- There are [tex]\( 6 \times 6 = 36 \)[/tex] total possible outcomes for two dice.
- Thus, [tex]\( P(S_6 | H) = \frac{6}{36} = \frac{1}{6} \)[/tex].
7. Probability of Scoring 6 Given Tail:
- There are 4 occurrences where the product of two dice is 6.
- There are [tex]\( 6 \times 6 = 36 \)[/tex] total possible outcomes for two dice.
- Thus, [tex]\( P(S_6 | T) = \frac{4}{36} = \frac{1}{9} \)[/tex].
8. Total Probability of Scoring 6:
- Use the law of total probability:
[tex]\[ P(S_6) = P(S_6 | H) \cdot P(H) + P(S_6 | T) \cdot P(T) \][/tex]
- Substituting the known values:
[tex]\[ P(S_6) = \left( \frac{1}{6} \cdot \frac{1}{3} \right) + \left( \frac{1}{9} \cdot \frac{2}{3} \right) = \frac{1}{18} + \frac{2}{27} = \frac{1}{18} + \frac{2}{27} = \frac{3}{54} + \frac{4}{54} = \frac{7}{54} \approx 0.12963 \][/tex]
9. Bayes' Theorem:
- To find [tex]\( P(H | S_6) \)[/tex], we use Bayes' theorem:
[tex]\[ P(H | S_6) = \frac{P(S_6 | H) \cdot P(H)}{P(S_6)} \][/tex]
- Substituting the values:
[tex]\[ P(H | S_6) = \frac{\left( \frac{1}{6} \right) \cdot \left( \frac{1}{3} \right)}{\frac{7}{54}} = \frac{\frac{1}{18}}{\frac{7}{54}} = \frac{1}{18} \cdot \frac{54}{7} = \frac{3}{7} \approx 0.42857 \][/tex]
Therefore, the probability that the coin showed a head given that Jamie scores 6 is approximately [tex]\(\boxed{0.42857}\)[/tex].
Your participation is crucial to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. IDNLearn.com is your reliable source for answers. We appreciate your visit and look forward to assisting you again soon.
