Expert: Ahmed Salami - 10/22/2004

Question
" A student that just graduated from high school has a 61% chance of being accepted to university A, a 71% chance of being accepted into university B and a 38% chance of being accepted to university C.

a) What is the probability of being accepted by all the universities?
b) What is the probability of not being accepted to any university?
c) What is the probability of being accepted only to university B?
d) What is the probability of being accepted by at least one university?"

Answer
Hi Dee,
If the probability of an event occuring is p, the probability of the event not occuring is 1-p. lets label the events of success at the universities A,B,C and the events of failure as A',B',C'.
The probability of successes are 0.61,0.71 and 0.38 respectively. Those of failures are 0.39,0.29 and 0.62 .
a)The probability of being accepted by all the universities can be written as
P(ABC)= P(A).P(B).P(C)
     = (0.61)(0.71)(0.38)
     = 0.1646 or 16.46%
b)The probability of not being accepted to any university can be written as
P(A'B'C')= P(A').P(B').P(C')
        = (0.39)(0.29)(0.62)
        = 0.0701 or 7.01%
c)The probability of being accepted only to university B can be written as
P(A'BC')= P(A').P(B).P(C')
       = (0.39)(0.71)(0.62)
       = 0.1717 or 17.17%
d)Being accepted by at least one university is the opposite of not being accepted by any university. Therefore, the probability of being accepted by at least one university can be written as
1 - P(A'B'C')
= 1 - 0.0701
= 0.9299 or 92.99%
But if you want to try out the longer method, just add the probabilities of the various ways of having at least one success. 'At least one success' means either one success (which could be any of the events AB'C',A'BC' or A'B'C) or two successes (which could be any of the events ABC',AB'C or A'BC)or three successes (ABC). Add all the probabilities of the seven possible events and you should get the same thing.

I hope you understand it. You can always get back to me.
Sorry for any delay.
Regards.