An IT worker works from home 1 day a week. When she works from home she answers 80% of the emails within an hour, 15% of the emails within 2 hours and 5% of the emails within the day. When she is at the office, she answers 50% of the emails within an hour, 40% within 2 hours and 10% within the day.
- Question 1: If you send her an email, what is the probability that she will answer within 2 hours?
- Question 2: Given that she hasn’t replied to your email within the first 2 hours, what is the probability that she is working from home?
I am trying to solve this question, but I am not sure how to do it.
I tried using a decision tree for working from home and away for both questions. I do not think it is the right way.
For the second question, if she has not replied within 2 hours, it means she will reply within a day, right?
$$P(Home|1 day) = \frac{P(1 day|Home) \times P(Home)}{P(1 day)} =$$
$$= (\frac{5}{100} \times \frac{1}{7}) / (\frac{1}{7} \times \frac{5}{100}) + \frac{6}{7} \times \frac{10}{100} = 0.076$$
?
Note that the problem is poorly worded. It doesn't specify whether you the sender can send your (initial) message anytime of day. If you send it in the morning you'll get different reply times than if you send in the last 5 minutes of the day. Thus I assume you the sender are sending your initial message at the beginning of the day.
I'm assuming five work days per week. I'm also assuming that if you get a reply in $30$ minutes, that is indeed "within two hours."
Problem 1
$$P(0 \to 2) = P(0 \to 1) + P(1 \to 2)$$
$$= (P(0 \to 1|h) + P(1 \to 2|h)) P(h) + (P(0 \to 1|o) + P(1 \to 2|o) P(o))$$
$$= (0.8 + 0.15) 0.2 + (0.5 + 0.4) .8 = 0.91.$$
Here $h$ denotes "home" and $o$ denotes "office".
Problem 2
$$\frac{P(2 \to 8|h) P(h)}{P(2 \to 8|h)P(h)+ P(2 \to 8|o)P(o)}$$
$$= \frac{(0.5)(0.2)}{(0.5)(0.2) + (0.1)(0.8)} = 0.2$$
where $2 \to 8$ denotes the reply comes within the (8-hour) day after two hours.