Venn Diagram: At a certain college it is known that 2/3....

[lea8802]

At a certain college it is known that 2/3 of the students are under 21 years of age. Also 3/5 of the students are female and 1/2 of all students are female and under 21 years of age.

What are the possibilities that they are male and over 21?
What are the possibilities that they are male or over 21?

 

[soroban]

Re: Venn Diagram

Hello, lea8802!

At a certain college it is known that \(\displaystyle \frac{2}{3}\) of the students are under 21 years of age.
Also \(\displaystyle \frac{3}{5}\)of the students are female
and \(\displaystyle \frac{1}{2}\) of all students are female and under 21 years of age.

What are the possibilities that they are male and over 21?
What are the possibilities that they are male or over 21?
. . Do you mean "probabiity"?

Make a chart . . .

          |  Female |  Male | Total |
- - - - - + - - - - + - - - + - - - +
 Under 21 |         |       |       |
- - - - - + - - - - + - - - + - - - +
  Over 21 |         |       |       |
- - - - - + - - - - + - - - + - - - +
   Total  |         |       |       |
- - - - - + - - - - + - - - + - - - +

We are told: .\(\displaystyle \frac{2}{3}\) are Under 21.
. . Hence, \(\displaystyle \frac{1}{3}\) are Over 21.

We are told: .\(\displaystyle \frac{3}{5}\) of the students are Female.
. . Hence, \(\displaystyle \frac{2}{5}\) of the students are Male.

Write those facts in the chart:

          |  Female |  Male | Total |
- - - - - + - - - - + - - - + - - - +
 Under 21 |   1/2   |       |  2/3  |
- - - - - + - - - - + - - - + - - - +
  Over 21 |  1/10   |       |  1/3  |
- - - - - + - - - - + - - - + - - - +
   Total  |   3/5   |  2/5  |   1   |
- - - - - + - - - - + - - - + - - - +

Since this is basically an "addition table",
. . the remaining cells can be filled in.

          |  Female |  Male | Total |
- - - - - + - - - - + - - - + - - - +
 Under 21 |   1/2   |   1/6 |  2/3  |
- - - - - + - - - - + - - - + - - - +
  Over 21 |  1/10   |  7/30 |  1/3  |
- - - - - + - - - - + - - - + - - - +
   Total  |   3/5   |  2/5  |   1   |
- - - - - + - - - - + - - - + - - - +

And now you can answer the questions . . .

 

[lea8802]

Venn Diagram

So male and over 21 would be 7/30 and male or over 21 would be 11/15?

 

[soroban]

Re: Venn Diagram

Hello again, lea8802!

So male and over 21 would be \(\displaystyle \frac{7}{30}\;\) . . . yes

and male or over 21 would be \(\displaystyle \frac{11}{15}\;\) . . . no

\(\displaystyle P(\text{male}\,\cup\,\text{over 21}) \:=\:\L\frac{1}{10}\,+\,\frac{7}{30} \,+\,\frac{1}{6}\;=\;\frac{3}{30}\,+\,\frac{7}{30}\,+\,\frac{5}{30}\;=\;\frac{15}{30} \;=\;\fbox{\frac{1}{2}}\)

 

[lea8802]

Thank you, you were a huge help!