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. .25*.35
b. .35*.75+.25*.65 + .25*.35
c.
d.
I will be happy to critique your thinking. In reality, the events are not independent, assuming the schools judge admission on nearly the same factors.
thx for your help, i had that for a but for b i use p(a) = 1-p(b) and for c i used P(a or b) = .25 + .35 = 60 and for d i caluculated that p(a): 1-.25 =.75 p(b): 1-.35 = .65 then i multiplied the two .75*.65=.4875 = 48.75%
I really appreciate your assistance
To solve this problem, we can use the concept of probability and apply it to each question.
a. P(accepted in both universities):
Since the events are independent, the probability of being accepted in both universities is the product of their individual acceptance probabilities.
P(Accepted in both) = P(Accepted in University A) * P(Accepted in University B)
P(Accepted in both) = 0.25 * 0.35
P(Accepted in both) = 0.0875
So, the probability that you will be accepted in both universities is 0.0875 or 8.75%.
b. P(accepted to at least one graduate program):
To find the probability of being accepted to at least one graduate program, we can calculate the complement of the probability of being accepted by both universities.
P(Accepted to at least one) = 1 - P(Not accepted in both)
P(Accepted to at least one) = 1 - P(Not accepted in University A) * P(Not accepted in University B)
Since the complement of being accepted is not being accepted, we can calculate the probability of not being accepted by subtracting the acceptance probability from 1.
P(Not accepted in University A) = 1 - P(Accepted in University A)
P(Not accepted in University A) = 1 - 0.25
P(Not accepted in University A) = 0.75
Similarly,
P(Not accepted in University B) = 1 - P(Accepted in University B)
P(Not accepted in University B) = 1 - 0.35
P(Not accepted in University B) = 0.65
Now we can calculate:
P(Accepted to at least one) = 1 - (0.75 * 0.65)
P(Accepted to at least one) = 1 - 0.4875
P(Accepted to at least one) = 0.5125
So, the probability that you will be accepted to at least one graduate program is 0.5125 or 51.25%.
c. P(one and only one university accepts you):
To find the probability that only one university will accept you, we need to consider the following cases:
1. Accepted in University A, Not accepted in University B:
P(Accepted in University A) * P(Not accepted in University B)
2. Not accepted in University A, Accepted in University B:
P(Not accepted in University A) * P(Accepted in University B)
We can sum up the probabilities of both cases to get the desired probability:
P(one and only one accepts you) = (P(Accepted in University A) * P(Not accepted in University B)) +
(P(Not accepted in University A) * P(Accepted in University B))
P(one and only one accepts you) = (0.25 * 0.65) + (0.75 * 0.35)
P(one and only one accepts you) = 0.1625 + 0.2625
P(one and only one accepts you) = 0.425
So, the probability that one and only one of the universities will accept you is 0.425 or 42.5%.
d. P(neither university accepts you):
To find the probability that neither university will accept you, we can multiply the probabilities of not being accepted in both universities.
P(Neither university accepts you) = P(Not accepted in University A) * P(Not accepted in University B)
P(Neither university accepts you) = 0.75 * 0.65
P(Neither university accepts you) = 0.4875
So, the probability that neither university will accept you is 0.4875 or 48.75%.