Assume you have applied to two different universities (lets refer to them as Universities A and B) for your graduate program. In the past 25% of students (with similar credentials as yours) who applied to University A were accepted, while University B accepted 35% of the applicants. Assume events are independent of each other.
a. probability that you will be accepted in both
b. probability that you will be accepted to at least one graduate program
c. probability that one and only one of the universities will accept you
d. probability that neither university will accept you
a.) You have to assume that the probability of being accepted with "similar credentials" is a random process. This may not be the case, however. Admissions committees do not roll dice. They may look for similar admission criteria that are not easily quantified. Assuming that it is random however, the probability of being accepted at both is (0.25)(0.35) = 0.0875
b.) This probability is 1 MINUS the probability of being rejected by both colleges. That equals 1 - (0.75)(0.65) = 0.5125
c.) Add the probability of acceptance by A and rejection by B to the probability of acceptance by B and rejection by A. That is
(0.75)(0.35) + (0.25)(0.65) = 0.425
d.) (0.75)(0.65) = 0.4875
thx that really helped, however for c i thought it was mutually exclusive so i added the two probabilities .25 and .35. i didn't realize that i needed to include the rejection percentage.
To answer these questions, we will use the concept of probability and the assumption that the events are independent of each other. Let's break down each question step by step:
a. Probability that you will be accepted in both:
Since the acceptance at each university is independent, we can multiply the probabilities of being accepted at each university.
Probability of being accepted at University A = 25% = 0.25
Probability of being accepted at University B = 35% = 0.35
Therefore, the probability of being accepted at both universities is:
Probability of being accepted at both = Probability of being accepted at A * Probability of being accepted at B
= 0.25 * 0.35
= 0.0875
So, the probability of being accepted at both universities is 0.0875 or 8.75%.
b. Probability that you will be accepted to at least one graduate program:
To calculate this probability, we need to find the complement of the event that you are not accepted at any of the universities and subtract it from 1.
Probability of not being accepted at University A = 1 - 0.25 = 0.75
Probability of not being accepted at University B = 1 - 0.35 = 0.65
Since the events of not being accepted at each university are independent, we can multiply their probabilities.
Probability of not being accepted at both universities = Probability of not being accepted at A * Probability of not being accepted at B
= 0.75 * 0.65
= 0.4875
Therefore, the probability of being accepted to at least one graduate program is:
Probability of being accepted to at least one = 1 - Probability of not being accepted at both
= 1 - 0.4875
= 0.5125
So, the probability of being accepted to at least one graduate program is 0.5125 or 51.25%.
c. Probability that one and only one of the universities will accept you:
There are two possibilities for this event: being accepted at University A but not at University B, or being accepted at University B but not at University A.
Probability of being accepted at University A and not at University B = Probability of being accepted at A * Probability of not being accepted at B
= 0.25 * 0.65
= 0.1625
Probability of being accepted at University B and not at University A = Probability of not being accepted at A * Probability of being accepted at B
= 0.75 * 0.35
= 0.2625
Therefore, the probability that one and only one of the universities will accept you is:
Probability of one and only one acceptance = Probability of being accepted at A but not at B + Probability of being accepted at B but not at A
= 0.1625 + 0.2625
= 0.425
So, the probability that one and only one of the universities will accept you is 0.425 or 42.5%.
d. Probability that neither university will accept you:
The probability that neither university will accept you is the complement of the event that you are accepted at at least one university.
Probability of neither acceptance = 1 - Probability of being accepted at least one
= 1 - 0.5125
= 0.4875
So, the probability that neither university will accept you is 0.4875 or 48.75%.
To solve these probabilities, we can use basic principles of probability and the given percentages. Let's break down each question step by step.
a. Probability that you will be accepted in both:
Since the events are independent, we can multiply the probabilities for each university.
P(A) = 0.25 (probability of acceptance at University A)
P(B) = 0.35 (probability of acceptance at University B)
P(A and B) = P(A) * P(B)
P(A and B) = 0.25 * 0.35
P(A and B) = 0.0875
So, the probability that you will be accepted at both universities is 0.0875 or 8.75%.
b. Probability that you will be accepted to at least one graduate program:
To calculate this probability, we need to consider the opposite event, which is the probability of not being accepted to any university. Then we can subtract it from 1 to find the probability of being accepted at least at one university.
P(not accepted at any) = (1 - P(A)) * (1 - P(B))
P(not accepted at any) = (1 - 0.25) * (1 - 0.35)
P(not accepted at any) = 0.75 * 0.65
P(not accepted at any) = 0.4875
P(accepted at least one) = 1 - P(not accepted at any)
P(accepted at least one) = 1 - 0.4875
P(accepted at least one) = 0.5125
So, the probability that you will be accepted at least at one university is 0.5125 or 51.25%.
c. Probability that one and only one of the universities will accept you:
For this probability, we need to consider two mutually exclusive events: being accepted by University A and not being accepted by University B, or being accepted by University B and not being accepted by University A. Then we can add these probabilities.
P(A and not B) = P(A) * (1 - P(B))
P(not A and B) = (1 - P(A)) * P(B)
P(one and only one of the universities accepts) = P(A and not B) + P(not A and B)
P(one and only one of the universities accepts) = (0.25 * (1 - 0.35)) + ((1 - 0.25) * 0.35)
P(one and only one of the universities accepts) = (0.25 * 0.65) + (0.75 * 0.35)
P(one and only one of the universities accepts) = 0.1625 + 0.2625
P(one and only one of the universities accepts) = 0.425
So, the probability that one and only one of the universities will accept you is 0.425 or 42.5%.
d. Probability that neither university will accept you:
This probability is simply the complement of the probability of being accepted in at least one university.
P(neither university accepts) = 1 - P(accepted at least one)
P(neither university accepts) = 1 - 0.5125
P(neither university accepts) = 0.4875
So, the probability that neither university will accept you is 0.4875 or 48.75%.