A father is questioning the probability his daughter will receive a scholarship. She is a finalist for two different scholarships. The awarding of these scholarships is independent. For the first scholarship (Scholarship A) there are 5 finalists and two scholarships will be awarded. For the second scholarship (Scholarship B) there are also five finalists and one scholarship will be awarded.
1. What is the probability that she will receive Scholarship A and Scholarship B?
2. What is the probability that she will receive at least one scholarship, Scholarship A or Scholarship B?
3. Should she apply for both scholarships or one of the scholarships? If just one, which one? Why?

So for the first scholarship, there are five finalist and only two scholarship will be awarded. And for the second scholarship there are five finalist and only one scholarship will be awarded.

Therefore,

[tex]$P(A)=\frac{2}{5}$[/tex] and [tex]$P(B)=\frac{1}{5}$[/tex]

1. So, [tex]$P(A \cap B ) = P(A) P(B)$[/tex]

[tex]$=\frac{2}{5} \times \frac{1}{5}$[/tex]

[tex]$=\frac{2}{25}$[/tex]

2. [tex]$P(A \cup B) = P(A)+P(B)-P(A)P(B)$[/tex]

[tex]$=\frac{2}{5}+\frac{1}{5}-\frac{2}{25}$[/tex]

[tex]$=\frac{13}{25}$[/tex]

3. She should apply to both the scholarships, so as to have a maximum chances of winning at least one scholarship.