Question 1205234
<font color=black size=3>
I'll use a two-way table instead of a tree diagram.


Here's the starting template
<table border = "1" cellpadding = "5"><tr><td></td><td>Male</td><td>Female</td><td>Total</td></tr><tr><td>Aid</td><td></td><td></td><td></td></tr><tr><td>No Aid</td><td></td><td></td><td></td></tr><tr><td>Total</td><td></td><td></td><td></td></tr></table>
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.
<table border = "1" cellpadding = "5"><tr><td></td><td>Male</td><td>Female</td><td>Total</td></tr><tr><td>Aid</td><td>4,892,238</td><td>7,158,960</td><td>12,051,198</td></tr><tr><td>No Aid</td><td></td><td></td><td></td></tr><tr><td>Total</td><td>8,073,000</td><td>10,980,000</td><td>19,053,000</td></tr></table>


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.


<table border = "1" cellpadding = "5"><tr><td></td><td>Male</td><td>Female</td><td>Total</td></tr><tr><td>Aid</td><td>4,892,238</td><td>7,158,960</td><td>12,051,198</td></tr><tr><td>No Aid</td><td>3,180,762</td><td>3,821,040</td><td>7,001,802</td></tr><tr><td>Total</td><td>8,073,000</td><td>10,980,000</td><td>19,053,000</td></tr></table>


Of those who got aid, we have this sub-table
<table border = "1" cellpadding = "5"><tr><td></td><td>Male</td><td>Female</td><td>Total</td></tr><tr><td>Federal</td><td>2,191,723</td><td>3,608,116</td><td>5,799,839</td></tr><tr><td>Not federal</td><td>2,700,515</td><td>3,550,844</td><td>6,251,359</td></tr><tr><td>Total</td><td>4,892,238</td><td>7,158,960</td><td>12,051,198</td></tr></table>
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.


-------------------------------------------------


Part (a)


Refer to the first table
<table border = "1" cellpadding = "5"><tr><td></td><td>Male</td><td>Female</td><td>Total</td></tr><tr><td>Aid</td><td>4,892,238</td><td>7,158,960</td><td>12,051,198</td></tr><tr><td>No Aid</td><td>3,180,762</td><td>3,821,040</td><td>7,001,802</td></tr><tr><td>Total</td><td>8,073,000</td><td>10,980,000</td><td>19,053,000</td></tr></table>


There are 3,180,762 men who did not get aid out of 19,053,000 students.


3180762/19053000 = <font color=red>0.166943</font> approximately


-------------------------------------------------


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 = <font color=red>0.405954</font> approximately

-------------------------------------------------


Part (c)


We'll need to use this table
<table border = "1" cellpadding = "5"><tr><td></td><td>Male</td><td>Female</td><td>Total</td></tr><tr><td>Aid</td><td>4,892,238</td><td>7,158,960</td><td>12,051,198</td></tr><tr><td>No Aid</td><td>3,180,762</td><td>3,821,040</td><td>7,001,802</td></tr><tr><td>Total</td><td>8,073,000</td><td>10,980,000</td><td>19,053,000</td></tr></table>
and this table as well
<table border = "1" cellpadding = "5"><tr><td></td><td>Male</td><td>Female</td><td>Total</td></tr><tr><td>Federal</td><td>2,191,723</td><td>3,608,116</td><td>5,799,839</td></tr><tr><td>Not federal</td><td>2,700,515</td><td>3,550,844</td><td>6,251,359</td></tr><tr><td>Total</td><td>4,892,238</td><td>7,158,960</td><td>12,051,198</td></tr></table>
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 = <font color=red>0.69132</font> approximately.

-----------------------------------------------------------
-----------------------------------------------------------


Summary
(a) <font color=red>0.166943</font>
(b) <font color=red>0.405954</font> 
(c) <font color=red>0.69132</font>
Each value is approximate. Round them however needed.
</font>