Can anyone help me set this up correctly, please:
John is going to eat at at McDonald's. The time of his arrival is uniformly distributed between 6PM and 7PM and it takes him 15 minutes to eat. Mary is also going to eat at McDonald's. The time of her arrival is uniformly distributed between 6:30PM and 7:15PM and it takes her 25 minutes to eat. Suppose the times of their two arrivals are independent of each other. What is the probability that there will be some time that they are both at McDonald's, i.e. their times at McDonald's overlap.
So let
$T$= John's arrival time
and
$S$=Mary's arrival time
I don't get how to set up the density function. For John, I got $f(t)=1/60$ for $0<t<60$. Is that correct? I think I can figure out the probability there is overlap by solving $Pr(T<S)$ and $Pr(S<T)$, but I am stuck on setting up the equations.
Thank you
John's arrival time is uniformly distributed on $[0,60]$ so we have $f_T(t)=\frac{1}{60}$ for $0\leq t \leq 60$, and Mary's arrival time is uniformly distributed on $[30,75]$ so $f_S(s)=\frac{1}{45}$ for $30\leq s \leq 75$. Since they arrive independently, their joint arrival time has density $$f_{TS}(t,s)=f_T(t)f_S(s)=\frac{1}{2700}$$
for $0 \leq t\leq 60,30\leq s\leq 75$.
They meet when one of two things happens: either Mary arrives less than 15 minutes after John, i.e., $S\leq T+15$; or John arrives less then 25 minutes after Mary, i.e. $T\leq S+25$. At this point it is extremely handy to draw a picture.
Hopefully you can sort out the rest.