Question 1205234: In a recent year 8,073,000 male students and 10,980,000 female students were
enrolled as undergraduates. Receiving aid were 60.6% of the male students and
65.2% of the female students. Of those receiving aid, 44.8% of the males got
federal aid and 50.4% of the females got federal aid. Choose 1 student at random.
(Hint: Make a tree diagram.) Find the probability that the student is
a. A male student without aid
b. A male student, given that the student has aid
c. A female student or a student who receives federal aid
Found 2 solutions by mccravyedwin, math_tutor2020:
Answer by mccravyedwin(408)   (Show Source):
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
I'll use a two-way table instead of a tree diagram.
Here's the starting template
| Male | Female | Total | | Aid | | | | | No Aid | | | | | Total | | | |
Place the values 8,073,000 and 10,980,000 at the bottom row of the male and female columns respectively.
60.6% of 8,073,000 = 0.606*(8,073,000) = 4,892,238 male students got aid.
65.2% of 10,980,000 = 0.652*(10,980,000) = 7,158,960 female students got aid.
Let's place those results into the table.
| Male | Female | Total | | Aid | 4,892,238 | 7,158,960 | 12,051,198 | | No Aid | | | | | Total | 8,073,000 | 10,980,000 | 19,053,000 |
To finish up the table, subtract the count who got aid from the total.
Eg: 8,073,000 - 4,892,238 = 3,180,762 male students did not get aid.
| Male | Female | Total | | Aid | 4,892,238 | 7,158,960 | 12,051,198 | | No Aid | 3,180,762 | 3,821,040 | 7,001,802 | | Total | 8,073,000 | 10,980,000 | 19,053,000 |
Of those who got aid, we have this sub-table
| Male | Female | Total | | Federal | 2,191,723 | 3,608,116 | 5,799,839 | | Not federal | 2,700,515 | 3,550,844 | 6,251,359 | | Total | 4,892,238 | 7,158,960 | 12,051,198 |
Example: 44.8% of 4,892,238 = 0.448*4892238 = 2191722.624 which rounds to 2191723
The values in the bottom row match up with the values in row 1 of the previous table. Spreadsheet software is strongly recommended.
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Part (a)
Refer to the first table
| Male | Female | Total | | Aid | 4,892,238 | 7,158,960 | 12,051,198 | | No Aid | 3,180,762 | 3,821,040 | 7,001,802 | | Total | 8,073,000 | 10,980,000 | 19,053,000 |
There are 3,180,762 men who did not get aid out of 19,053,000 students.
3180762/19053000 = 0.166943 approximately
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Part (b)
Refer to the same table as mentioned in part (a)
The key term "given" means we know 100% that whoever is randomly selected, they received aid of some kind (federal or non-federal).
We focus entirely on the "aid" row.
There are 4,892,238 men who got aid out of 12,051,198 students who got aid.
4892238/12051198 = 0.405954 approximately
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Part (c)
We'll need to use this table
| Male | Female | Total | | Aid | 4,892,238 | 7,158,960 | 12,051,198 | | No Aid | 3,180,762 | 3,821,040 | 7,001,802 | | Total | 8,073,000 | 10,980,000 | 19,053,000 |
and this table as well
| Male | Female | Total | | Federal | 2,191,723 | 3,608,116 | 5,799,839 | | Not federal | 2,700,515 | 3,550,844 | 6,251,359 | | Total | 4,892,238 | 7,158,960 | 12,051,198 |
There's probably a clever way to combine the two tables, but my mind is drawing a blank right now.
There are 10,980,000 female students and 5,799,839 students who got federal aid. The overlap is the 3,608,116 female students who got federal aid.
Use the inclusion exclusion principle to say that there are
10980000+5799839-3608116 = 13171723
students who are female, got federal aid, or both.
This is out of the 19053000 students total.
13171723/19053000 = 0.69132 approximately.
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Summary
(a) 0.166943
(b) 0.405954
(c) 0.69132
Each value is approximate. Round them however needed.
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