Given:
The total number of people are T = 267.
Explanation:
(a)
Determine the total number of teens.
![\begin{gathered} t=39+43+25 \\ =107 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20t%3D39%2B43%2B25%20%5C%5C%20%3D107%20%5Cend%7Bgathered%7D)
The probability for the person to be a teen is,
![\begin{gathered} P(t)=\frac{107}{267} \\ =0.4007 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20P%28t%29%3D%5Cfrac%7B107%7D%7B267%7D%20%5C%5C%20%3D0.4007%20%5Cend%7Bgathered%7D)
(b)
Determine the number of people like the vanilla.
![\begin{gathered} v=22+43+44 \\ =109 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20v%3D22%2B43%2B44%20%5C%5C%20%3D109%20%5Cend%7Bgathered%7D)
Determine the propability for the person selected likes the Vanilla.
![\begin{gathered} P(v)=\frac{109}{267} \\ =0.4082 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20P%28v%29%3D%5Cfrac%7B109%7D%7B267%7D%20%5C%5C%20%3D0.4082%20%5Cend%7Bgathered%7D)
(c)
Determine the number of people that are teen or likes the vanailla.
![\begin{gathered} v\text{ OR t=}22+43+44+39+25 \\ =173 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20v%5Ctext%7B%20OR%20t%3D%7D22%2B43%2B44%2B39%2B25%20%5C%5C%20%3D173%20%5Cend%7Bgathered%7D)
Determine the probability for a selected person is a teen or like the vanilla.
![\begin{gathered} P\text{ (v or t)=}\frac{173}{267} \\ =0.6479 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20P%5Ctext%7B%20%28v%20or%20t%29%3D%7D%5Cfrac%7B173%7D%7B267%7D%20%5C%5C%20%3D0.6479%20%5Cend%7Bgathered%7D)
(d)
The number of person that is teen and likes strawberry best are 25.
Determine the probability for selected person is teen and likes strawberry best.
![\begin{gathered} P\text{ (t and s)=}\frac{25}{267} \\ =0.0936 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20P%5Ctext%7B%20%28t%20and%20s%29%3D%7D%5Cfrac%7B25%7D%7B267%7D%20%5C%5C%20%3D0.0936%20%5Cend%7Bgathered%7D)
(e)
Determine the number of persons that are children.
![\begin{gathered} C=21+22+9 \\ =52 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20C%3D21%2B22%2B9%20%5C%5C%20%3D52%20%5Cend%7Bgathered%7D)
Determine the number person that are children and likes strawberry best.
![\text{ C and s=}9](https://tex.z-dn.net/?f=%5Ctext%7B%20C%20and%20s%3D%7D9)
So probability for the person likes the strawbwerry given that person is children is,
![\begin{gathered} P\text{ (s|C)=}\frac{P\text{ (s and c)}}{P\text{ (c)}} \\ =\frac{\frac{9}{267}}{\frac{52}{267}} \\ =\frac{9}{52} \\ \approx0.1731 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20P%5Ctext%7B%20%28s%7CC%29%3D%7D%5Cfrac%7BP%5Ctext%7B%20%28s%20and%20c%29%7D%7D%7BP%5Ctext%7B%20%28c%29%7D%7D%20%5C%5C%20%3D%5Cfrac%7B%5Cfrac%7B9%7D%7B267%7D%7D%7B%5Cfrac%7B52%7D%7B267%7D%7D%20%5C%5C%20%3D%5Cfrac%7B9%7D%7B52%7D%20%5C%5C%20%5Capprox0.1731%20%5Cend%7Bgathered%7D)
(f)
The number of persons who likes the strawberry best is,
![\begin{gathered} s=9+25+25 \\ =62 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20s%3D9%2B25%2B25%20%5C%5C%20%3D62%20%5Cend%7Bgathered%7D)
Determine the probability for person is teen given that person likes strawberry best.
![\begin{gathered} P\text{ (}t|s\text{)}=\frac{P\text{ (t and s)}}{P(s)} \\ =\frac{\frac{25}{267}}{\frac{62}{267}} \\ =\frac{25}{62} \\ =0.4032 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20P%5Ctext%7B%20%28%7Dt%7Cs%5Ctext%7B%29%7D%3D%5Cfrac%7BP%5Ctext%7B%20%28t%20and%20s%29%7D%7D%7BP%28s%29%7D%20%5C%5C%20%3D%5Cfrac%7B%5Cfrac%7B25%7D%7B267%7D%7D%7B%5Cfrac%7B62%7D%7B267%7D%7D%20%5C%5C%20%3D%5Cfrac%7B25%7D%7B62%7D%20%5C%5C%20%3D0.4032%20%5Cend%7Bgathered%7D)
Answer:
Is a teen: 0.4007
Likes vanilla the best: 0.4082
Is a teen or likes the Vanilla the best: 0.6479
Is a teen and likes the strawberry best: 0.0936
Likes strawberry the best GIVEN the person is child: 0.1732
Is a teen GIVEN that person likes the strawberry best: 0.4032