Question

A student is applying to two different agencies for scholarships. Based on the student’s academic record, the probability that the student will be awarded a scholarship from Agency A is 0.55 and the probability that the student will be awarded a scholarship from Agency B is 0.40. Furthermore, if the student is awarded a scholarship from Agency A, the probability that the student will be awarded a scholarship from Agency B is 0.60. What is the probability that the student will be awarded at least one of the two scholarships?

Answer 1

Answer:

There is a 62% probability that the student will be awarded at least one of the two scholarships.

Step-by-step explanation:

We solve this problem building the Venn's diagram of these probabilities.

I am going to say that:

A is the probability that the student gets a scolarship from Agency A.

B is the probability that the student gets a scolarship from Agency B.

We have that:

$A = a + (A \cap B)$

In which a is the probability that the student will get an scolarship from agency A but not from agency B and $A \cap B$ is the probability that the student will get an scolarship from both agencies.

By the same logic, we have that:

$B = b + (A \cap B)$

What is the probability that the student will be awarded at least one of the two scholarships?

This is

$P = a + b + (A \cap B)$

We have that:

$A = 0.55, B = 0.40$

If the student is awarded a scholarship from Agency A, the probability that the student will be awarded a scholarship from Agency B is 0.60.

This means that:

$\frac{A \cap B}{A} = 0.6$

$A \cap B = 0.6A = 0.6*0.55 = 0.33$

----------

$A = a + (A \cap B)$

$0.55 = a + 0.33$

$a = 0.22$

--------

$B = b + (A \cap B)$

$0.40 = b + 0.33$

$b = 0.07$

Answer:

$P = a + b + (A \cap B) = 0.22 + 0.07 + 0.33 = 0.62$

There is a 62% probability that the student will be awarded at least one of the two scholarships.