PROMPT: In desperate need of an ego boost, you turn to Tinder.

Assume for simplicity that arrival times of swipes are independent and identically

distributed. Let T be the time that a given person sees your profile. T is distributed as a

lifetime with

h(s) =

1 for s contained in [0; 1) [ [19; 24)

1/5 for s contained in [1; 19)

for s 2 [0; 24), and extended in the natural way for a multi-day period. In other words,

people are more likely to be on Tinder between 7PM and 1AM, than the rest of the day.

Once somebody comes upon your profile, they swipe right with probability 1/20.

(a) At 9PM, you come across a particularly desirable member of your preferred gender,

and naturally you swipe right on them. Not understanding how Tinder works, you

stare at your phone, waiting for them to swipe back. What is the probability that

you hear from the apple of your eye within one hour?

(b) You hear nothing, and it's 10PM. In a t of desperation, you open Tinder and swipe

right on 9 additional prospects. You go to bed, but put your phone on max volume

so that any notification from Tinder will wake you up. Uninterrupted, you would

sleep for 10 hours. Otherwise you wake up when you get your first match. Let S be

the amount of time you sleep. Find P(S > t) for t 2 (0;1).

(c) How long do you expect to sleep for? (You can set up the integral and use Mathe-

matica or any other software of your choice.)

I (believe) I have solved part a by finding P(T<21+1 | T>21) where T=21 represents 9pm. So part a should be good to go. How do I tackle parts b and c?

Thank you.