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Let's go through the problem step by step using the given data:
The personnel department analyzed 200 accountants, breaking down the count by their ages and their degrees:
- Under 30: 90 have BA degrees, 10 have MA degrees.
- 30 to 40: 20 have BA degrees, 30 have MA degrees.
- Over 40: 40 have BA degrees, 10 have MA degrees.
Now, we will solve the three parts A, B, and C as described in the question.
### A. The probability he has only a BA degree
To find the probability that a randomly selected accountant has only a BA degree, we first need the total number of accountants with BA degrees, and then divide it by the total number of accountants.
- The total number of accountants with BA degrees is:
[tex]\[ 90 \,(\text{Under 30}) + 20 \,(\text{30 to 40}) + 40 \,(\text{Over 40}) = 150 \][/tex]
- Thus, the probability that a randomly selected accountant has only a BA degree is:
[tex]\[ \text{Probability} = \frac{\text{Total number of BA}}{\text{Total number of accountants}} = \frac{150}{200} = 0.75 \][/tex]
### B. The probability he has an MA degree, given that he is over 40
To find the conditional probability that an accountant has an MA degree given that he is over 40, we need the number of accountants over 40 and the number of those over 40 with an MA degree:
- The total number of accountants over 40 is:
[tex]\[ 40 \,(\text{BA}) + 10 \,(\text{MA}) = 50 \][/tex]
- The number of accountants over 40 with an MA degree is 10.
- Thus, the probability that an accountant over 40 has an MA degree is:
[tex]\[ \text{Probability} = \frac{\text{Number of over 40 with MA}}{\text{Total number of over 40}} = \frac{10}{50} = 0.2 \][/tex]
### C. The probability he is under 30, given that he has a BA degree
To find the conditional probability that an accountant is under 30 given that he has a BA degree, we need the number of accountants with BA degrees and the number of those with BA who are under 30:
- The total number of accountants with BA degrees is 150 (as calculated in part A).
- The number of accountants with BA degrees who are under 30 is 90.
- Thus, the probability that an accountant with a BA degree is under 30 is:
[tex]\[ \text{Probability} = \frac{\text{Number of BA under 30}}{\text{Total number of BA}} = \frac{90}{150} = 0.6 \][/tex]
### Summary of Results
- A. The probability he has only a BA degree is [tex]\( \boxed{0.75} \)[/tex].
- B. The probability he has an MA degree, given that he is over 40 is [tex]\( \boxed{0.2} \)[/tex].
- C. The probability he is under 30, given that he has a BA degree is [tex]\( \boxed{0.6} \)[/tex].
The personnel department analyzed 200 accountants, breaking down the count by their ages and their degrees:
- Under 30: 90 have BA degrees, 10 have MA degrees.
- 30 to 40: 20 have BA degrees, 30 have MA degrees.
- Over 40: 40 have BA degrees, 10 have MA degrees.
Now, we will solve the three parts A, B, and C as described in the question.
### A. The probability he has only a BA degree
To find the probability that a randomly selected accountant has only a BA degree, we first need the total number of accountants with BA degrees, and then divide it by the total number of accountants.
- The total number of accountants with BA degrees is:
[tex]\[ 90 \,(\text{Under 30}) + 20 \,(\text{30 to 40}) + 40 \,(\text{Over 40}) = 150 \][/tex]
- Thus, the probability that a randomly selected accountant has only a BA degree is:
[tex]\[ \text{Probability} = \frac{\text{Total number of BA}}{\text{Total number of accountants}} = \frac{150}{200} = 0.75 \][/tex]
### B. The probability he has an MA degree, given that he is over 40
To find the conditional probability that an accountant has an MA degree given that he is over 40, we need the number of accountants over 40 and the number of those over 40 with an MA degree:
- The total number of accountants over 40 is:
[tex]\[ 40 \,(\text{BA}) + 10 \,(\text{MA}) = 50 \][/tex]
- The number of accountants over 40 with an MA degree is 10.
- Thus, the probability that an accountant over 40 has an MA degree is:
[tex]\[ \text{Probability} = \frac{\text{Number of over 40 with MA}}{\text{Total number of over 40}} = \frac{10}{50} = 0.2 \][/tex]
### C. The probability he is under 30, given that he has a BA degree
To find the conditional probability that an accountant is under 30 given that he has a BA degree, we need the number of accountants with BA degrees and the number of those with BA who are under 30:
- The total number of accountants with BA degrees is 150 (as calculated in part A).
- The number of accountants with BA degrees who are under 30 is 90.
- Thus, the probability that an accountant with a BA degree is under 30 is:
[tex]\[ \text{Probability} = \frac{\text{Number of BA under 30}}{\text{Total number of BA}} = \frac{90}{150} = 0.6 \][/tex]
### Summary of Results
- A. The probability he has only a BA degree is [tex]\( \boxed{0.75} \)[/tex].
- B. The probability he has an MA degree, given that he is over 40 is [tex]\( \boxed{0.2} \)[/tex].
- C. The probability he is under 30, given that he has a BA degree is [tex]\( \boxed{0.6} \)[/tex].
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