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Answer :
To evaluate if the probability model is valid, we need to check two main conditions:
1. The Sum of Probabilities: All probabilities in a valid probability model should add up to 1. This indicates that one of the outcomes from the list will definitely occur.
2. Probabilities Range: Each probability should be a number between 0 and 1, inclusive. This means no probability should be negative or greater than 1.
Let's look at the probability values given for each subject:
- AP Stats: 0.62
- AP Lang.: 0.58
- AP Comp. Sci.: 0.31
- AP Euro.: 0.65
Step 1: Check the Sum of Probabilities
First, add up all the probabilities:
[tex]\[
0.62 + 0.58 + 0.31 + 0.65 = 2.16
\][/tex]
Since the sum of the probabilities is 2.16, which is not equal to 1, this does not satisfy the requirement for the sum to be 1.
Step 2: Check Each Probability's Range
Next, verify that each probability is between 0 and 1:
- 0.62 is between 0 and 1.
- 0.58 is between 0 and 1.
- 0.31 is between 0 and 1.
- 0.65 is between 0 and 1.
All the probabilities are within the valid range.
Conclusion
Based on the checks:
- The model is not valid because the sum of the probabilities is not equal to 1.
- However, the model is considered valid concerning the range since all probabilities fall between 0 and 1.
Therefore, the correct statement regarding the probability model is:
"The probability model is not valid because the sum of the probabilities is not 1." However, it is noted that "The probability model is valid because all of the probabilities are between 0 and 1."
1. The Sum of Probabilities: All probabilities in a valid probability model should add up to 1. This indicates that one of the outcomes from the list will definitely occur.
2. Probabilities Range: Each probability should be a number between 0 and 1, inclusive. This means no probability should be negative or greater than 1.
Let's look at the probability values given for each subject:
- AP Stats: 0.62
- AP Lang.: 0.58
- AP Comp. Sci.: 0.31
- AP Euro.: 0.65
Step 1: Check the Sum of Probabilities
First, add up all the probabilities:
[tex]\[
0.62 + 0.58 + 0.31 + 0.65 = 2.16
\][/tex]
Since the sum of the probabilities is 2.16, which is not equal to 1, this does not satisfy the requirement for the sum to be 1.
Step 2: Check Each Probability's Range
Next, verify that each probability is between 0 and 1:
- 0.62 is between 0 and 1.
- 0.58 is between 0 and 1.
- 0.31 is between 0 and 1.
- 0.65 is between 0 and 1.
All the probabilities are within the valid range.
Conclusion
Based on the checks:
- The model is not valid because the sum of the probabilities is not equal to 1.
- However, the model is considered valid concerning the range since all probabilities fall between 0 and 1.
Therefore, the correct statement regarding the probability model is:
"The probability model is not valid because the sum of the probabilities is not 1." However, it is noted that "The probability model is valid because all of the probabilities are between 0 and 1."
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