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?

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

By the same logic, we have that:

[tex]B = b + (A \cap B)[/tex]

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

This is

[tex]P = a + b + (A \cap B)[/tex]

We have that:

[tex]A = 0.55, B = 0.40[/tex]

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:

[tex]\frac{A \cap B}{A} = 0.6[/tex]

[tex]A \cap B = 0.6A = 0.6*0.55 = 0.33[/tex]

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[tex]A = a + (A \cap B)[/tex]

[tex]0.55 = a + 0.33[/tex]

[tex]a = 0.22[/tex]

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[tex]B = b + (A \cap B)[/tex]

[tex]0.40 = b + 0.33[/tex]

[tex]b = 0.07[/tex]

Answer:

[tex]P = a + b + (A \cap B) = 0.22 + 0.07 + 0.33 = 0.62[/tex]

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

MrRoyal MrRoyal · Answer 2 · 2021-10-18T12:29:07+02:00

Probabilities are used to determine the chances of an event.

The probability that the student will be awarded at least one of the two scholarships is 0.35

The given parameters are:

[tex]\mathbf{P(A) = 0.55}[/tex]

[tex]\mathbf{P(B) = 0.40}[/tex]

[tex]\mathbf{P(A\ and\ B) = 0.60}[/tex]

The probability that the student will be awarded at least one of the two scholarships is P(A or B)

So, we have:

[tex]\mathbf{P(A\ or\ B) = P(A) + P(B) - P(A\ and\ B)}[/tex]

This gives

[tex]\mathbf{P(A\ or\ B) = 0.55 + 0.40 - 0.60}[/tex]

[tex]\mathbf{P(A\ or\ B) = 0.35}[/tex]

Hence, probability that the student will be awarded at least one of the two scholarships is 0.35