Question

Alice and Bob visit the gym on Saturday for their hourlong workouts. Alice always arrives between 2:00 and 3:00 and Bob between 2:30 and 3:30. Assuming their arrival times are drawn independently and uniformly from the specified intervals (i.e. any arrival time in the given window is equally likely), and assuming they each stay for precisely one hour, what is the probability that on any given Saturday there exists a moment in time when all two are present at the gym? Please express your answer as a fraction in lowest terms.

Answer 1

let X is timing of Alice while Y is timing of Bob

so

f(x) =f(y) =1, 0<x<1,0<y<1

as X and Y are indepedent so f(x,y) =f(x)*f(y) =1

both will present in the gym under two scenerio

1) if Alice comes at anytime between 2:00-3:00 but Bob will come between 2:30-3:0 that is 0<x<1 and 0<y<0.5

or

2) alice comes in interval 2:30-3:00 and Bob can come at anytime in his interval 2:30-3:30 that 0.5<x<1 and 0<y<1

but there is common period in both scenerio that is 2:30-3:00 that is 0.5<x<1,0<y<0.5

so we have to find P[(0<X<1,0<y<0.5) or  (0.5<X<1,0<Y<1)]

now required probability