Question 1072401
p(A) = 3/4


p(B) = 4/5


since these are independent events, .....


p(A and B) = p(A) * p(B) = 3/4 * 4/5 = 12/20 = 3/5.


p(A or B) = p(A) + p(B) - p(A and B)


p(A or B) is the probability that the problem is solved, since either A solves it or B solves it or both A and B solve it at the same time.


p(A or B) = p(A) + p(B) - p(A and B) becomes:


p(A or B) = 3/4 + 4/5 - 3/5.


put everything under the same common denominator and you get:


p(A or B) = 15/20 + 16/20 - 12/20


combine like terms under the same denominator and you get:


p(A or B) = (15 + 16 - 12) / 20


combine like terms to get:


p(A or B) = 19 / 20.


that's the probability that the problem will be solved.


the probability that the problem will not be solved is 1 - 19/20 = 1/20.


in the formula of p(A or B) = p(A) + p(B) - p(A and B), p(A and B) is assumed to be part of p(A) and part of p(B) which means they are being double counted.


the subtraction of p(A and B) therefore eliminates the double counting.