Question

You’re ordering raw material from two different suppliers (A, B) to produce the chassis of an unmanned aerial vehicle. Let ? denote the event that supplies from A arrive on time, and let B denote the event that supplies from ? arrive on time. Assume that A and B are independent and that ?(?) > ?(?). If ?(? ∪ ?) = 0.626 and ?(? ∩ ?) = 0.144, determine the values of ?(?) and ?(?).

Answer 1

?(? ∪ ?) = ?(?) + ?(?) - ?(? ∩ ?)

given ?(? ∩ ?) = 0.144, ?(? ∪ ?) = 0.626

Let P(A) = x, P(B) = y

Given A and B are independent events and hence ?(? ∩ ?) = P(A) * P(B)

0.144 = x*y

x = 0.144/y ---->(1)

?(? ∪ ?) = x + y -x*y

0.626 = x + y - 0.144

x + y = 0.77

0.144/y + y = 0.77

by rearranging the terms we get a quadratic equation as below

y² -0.77*y + 0.144 = 0

by solving this we get roots as y = 0.45, y = 0.32

for y = 0.45 , x = 0.77 - 0.45 = 0.32

for y = 0.32 , x = 0.77 - 0.32 = 0.45

given that ?(?) > ?(?).

hence the solution will be x = 0.45 , y = 0.32

?(?) = 0.45 and ?(?) = 0.32