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Answer :
To determine the validity of the probability model provided, let's break down the steps involved:
1. Probabilities Listed: We're given probabilities for four subjects:
- AP Stats: 0.62
- AP Lang: 0.58
- AP Comp. Sci.: 0.31
- AP Euro: 0.65
2. Sum of Probabilities: We need to check if the sum of these probabilities equals 1. If the sum is not equal to 1, the model is not valid in representing a complete probability distribution.
3. Calculate the Total:
- Add the probabilities: [tex]\(0.62 + 0.58 + 0.31 + 0.65 = 2.16\)[/tex].
4. Evaluate the Validity:
- For a probability model to be valid, the total sum should be exactly 1. In this case, the sum is 2.16, which is not equal to 1.
- Therefore, the statement "The probability model is valid because the sum of the probabilities is 1" is incorrect.
5. Check Each Probability:
- All probabilities are between 0 and 1, which is a necessary condition for a valid probability model. Thus, the statement "The probability model is valid because all of the probabilities are between 0 and 1" is true, but not sufficient on its own to determine the model's validity.
6. Conclusion:
- Since the sum of the probabilities is not equal to 1, the correct answer is: "The probability model is not valid because the sum of the probabilities is not 1."
Remember, for any probability model to be valid, the total of all probabilities must be 1, and each individual probability must fall between 0 and 1. In this scenario, while each probability is individually valid, the sum of all probabilities does not meet the requirement for a valid model.
1. Probabilities Listed: We're given probabilities for four subjects:
- AP Stats: 0.62
- AP Lang: 0.58
- AP Comp. Sci.: 0.31
- AP Euro: 0.65
2. Sum of Probabilities: We need to check if the sum of these probabilities equals 1. If the sum is not equal to 1, the model is not valid in representing a complete probability distribution.
3. Calculate the Total:
- Add the probabilities: [tex]\(0.62 + 0.58 + 0.31 + 0.65 = 2.16\)[/tex].
4. Evaluate the Validity:
- For a probability model to be valid, the total sum should be exactly 1. In this case, the sum is 2.16, which is not equal to 1.
- Therefore, the statement "The probability model is valid because the sum of the probabilities is 1" is incorrect.
5. Check Each Probability:
- All probabilities are between 0 and 1, which is a necessary condition for a valid probability model. Thus, the statement "The probability model is valid because all of the probabilities are between 0 and 1" is true, but not sufficient on its own to determine the model's validity.
6. Conclusion:
- Since the sum of the probabilities is not equal to 1, the correct answer is: "The probability model is not valid because the sum of the probabilities is not 1."
Remember, for any probability model to be valid, the total of all probabilities must be 1, and each individual probability must fall between 0 and 1. In this scenario, while each probability is individually valid, the sum of all probabilities does not meet the requirement for a valid model.
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