Student Financial Aid In a recent year 8,073,000
male students and 10,980,000 female students were
enrolled as undergraduates. Receiving aid were 60.6%
of the male students and 65.2% of the female students.
Of those receiving aid, 44.8% of the males got federal
aid and 50.4% of the females got federal aid. Choose 1
student at random. (Hint: Make a tree diagram.) Find
the probability that the student is
a. A male student without aid
b. A male student, given that the student has
aid
c. A female student or a student who receives federal
aid
a. A male student without aid:
To find the probability that a student is a male student without aid, we need to calculate the probability that a student is a male student who does not receive aid.
Given:
Total male students: 8,073,000
Total female students: 10,980,000
Probability of receiving aid for male students: 60.6%
Probability of receiving aid for female students: 65.2%
Probability of a male student without aid = 1 - Probability of a male student receiving aid
Probability of a male student receiving aid = Probability of receiving aid for male students = 60.6%
Probability of a male student without aid = 1 - 0.606 = 0.394
Therefore, the probability that a student is a male student without aid is 0.394 or 39.4%.
b. A male student, given that the student has aid:
To find the probability that a student is a male student, given that the student has aid, we need to calculate the conditional probability.
Probability of being a male student given that the student has aid = (Probability of a male student receiving aid) / (Probability of receiving aid for both males and females)
Probability of receiving aid for both males and females = (Probability of receiving aid for male students * Probability of being a male student) + (Probability of receiving aid for female students * Probability of being a female student)
Probability of receiving aid for both males and females = (0.606 * (8,073,000 / (8,073,000 + 10,980,000))) + (0.652 * (10,980,000 / (8,073,000 + 10,980,000)))
Probability of receiving aid for both males and females = 0.628
Probability of being a male student given that the student has aid = (0.606) / (0.628) ≈ 0.966 or 96.6%
Therefore, the probability that a student is a male student, given that the student has aid, is approximately 0.966 or 96.6%.
c. A female student or a student who receives federal aid:
To find the probability that a student is a female student or a student who receives federal aid, we need to calculate the probability of the union of these two events.
Probability of female students = Probability of receiving aid for female students = 65.2%
Probability of receiving federal aid for both males and females = (Probability of receiving aid for male students * Probability of being a male student) + (Probability of receiving aid for female students * Probability of being a female student)
Probability of receiving federal aid for both males and females = (0.448 * (8,073,000 / (8,073,000 + 10,980,000))) + (0.504 * (10,980,000 / (8,073,000 + 10,980,000)))
Probability of receiving federal aid for both males and females = 0.474
Probability of female student or a student who receives federal aid = Probability of female students + Probability of receiving federal aid for both males and females
Probability of female student or a student who receives federal aid = 0.652 + 0.474 = 1.126 (Not possible, as probability cannot be greater than 1)
Since there seems to be an error in the given values, the probability of a female student or a student who receives federal aid cannot be accurately determined with the given information.
To solve this problem, we can create a tree diagram that shows the different possibilities and probabilities at each step.
First, let's define some probabilities:
- P(M) = probability of selecting a male student
- P(F) = probability of selecting a female student
- P(A|M) = probability of a male student receiving aid
- P(A|F) = probability of a female student receiving aid
- P(Fed|M) = probability of a male student receiving federal aid
- P(Fed|F) = probability of a female student receiving federal aid
a. Probability of selecting a male student without aid:
This is equal to P(M) * (1 - P(A|M)). Using the information given, we have:
P(M) = 8,073,000 / (8,073,000 + 10,980,000) = 0.423
P(A|M) = 0.606
Therefore, the probability of selecting a male student without aid is:
P(M without aid) = P(M) * (1 - P(A|M)) = 0.423 * (1 - 0.606) ≈ 0.165
b. Probability of selecting a male student, given that the student has aid:
This is equal to P(M|A), which can be calculated using Bayes' theorem:
P(M|A) = (P(M) * P(A|M)) / P(A)
To find P(A), we need to consider the probability of any student receiving aid, whether they are male or female. This can be calculated as:
P(A) = P(A|M) * P(M) + P(A|F) * P(F)
P(M) and P(F) can be calculated as in part a:
P(M) = 0.423
P(F) = 1 - P(M) = 1 - 0.423 ≈ 0.577
P(A|M) and P(A|F) are given in the problem:
P(A|M) = 0.606
P(A|F) = 0.652
Using these probabilities, we can calculate P(A):
P(A) = P(A|M) * P(M) + P(A|F) * P(F) = 0.606 * 0.423 + 0.652 * 0.577 ≈ 0.526
Now we can calculate P(M|A):
P(M|A) = (P(M) * P(A|M)) / P(A) = (0.423 * 0.606) / 0.526 ≈ 0.488
c. Probability of selecting a female student or a student who receives federal aid:
This is equal to P(F) + P(Fed). P(F) is the probability of selecting a female student and can be calculated as:
P(F) = 1 - P(M) = 1 - 0.423 ≈ 0.577
P(Fed) is the probability of selecting a student who receives federal aid, regardless of gender. This can be calculated as:
P(Fed) = P(Fed|M) * P(M) + P(Fed|F) * P(F)
P(Fed|M) and P(Fed|F) are given in the problem:
P(Fed|M) = 0.448
P(Fed|F) = 0.504
Using these probabilities, we can calculate P(Fed):
P(Fed) = P(Fed|M) * P(M) + P(Fed|F) * P(F) = 0.448 * 0.423 + 0.504 * 0.577 ≈ 0.496
Therefore, the probability of selecting a female student or a student who receives federal aid is:
P(F or Fed) = P(F) + P(Fed) = 0.577 + 0.496 ≈ 1.073 (Note: Since probabilities cannot exceed 1, the total probability should be capped at 1)
To find the probabilities in this scenario, we can start by creating a tree diagram to represent all the possible outcomes.
1. Male student without aid:
From the given data, we know that 60.6% of male students receive aid, which means 100% - 60.6% = 39.4% of male students do not receive aid.
2. Male student with aid:
Out of the male students who receive aid, we know that 44.8% of them receive federal aid. Therefore, the probability of a male student having aid is 60.6% * 44.8% = 27.2%.
3. Female student without aid:
Following a similar pattern as above, we can calculate the probability of a female student not receiving aid: 100% - 65.2% = 34.8%.
4. Female student with aid:
Using the given data, we can find the probability of a female student receiving aid: 65.2% * 50.4% = 32.9%.
Now, let's compute the desired probabilities:
a. The probability that the student is a male student without aid is 39.4%.
b. The probability that the student is a male student, given that the student has aid is calculated as the probability of a male student with aid (27.2%) divided by the overall probability of having aid, which is the sum of male students with aid (27.2%) and female students with aid (32.9%): 27.2% / (27.2% + 32.9%) ≈ 45.27%.
c. The probability that the student is a female student or a student who receives federal aid can be found by considering the complement (not receiving aid) and the intersection (receiving federal aid) of these events. So, we add the probabilities of a female student without aid (34.8%) and a student who receives federal aid (27.2% + 32.9%): 34.8% + (27.2% + 32.9%) ≈ 94.9%.
Therefore:
a. The probability that the student is a male student without aid is 39.4%.
b. The probability that the student is a male student, given that the student has aid is around 45.27%.
c. The probability that the student is a female student or a student who receives federal aid is approximately 94.9%.