The Question:
Suppose that you receive emails at an average rate of 6 per hour. At 10:00am, your inbox is empty. At 10:10am, you receive a message on your computer that you have received at least one email. However, you know that this message is correct only 80% of the time (sometimes it tells you that you have received an email when, in fact, your inbox is still empty). What is the probability that you have received at least one email by 10:10am, given that you have received the message?
My thoughts:
Since there could be a continuous amount of variables in this window of opportunity, I believe it is a Poisson distribution. With the average being 6 per hour, the lambda for the formula should be 1 (EDIT: Correction from 6). This would give me the probability of 63.21205588%.
So I believe that we can find the probability of it being in the first 10 minutes by doing 1-Probability(No emails in 10 minutes) to give us an answer, 63.21205588%. Once we find that we can just do 63.21205588% * 0.8 since there is a 80% chance of it being correct. However I'm not sure if we need to calculate 0.2 multiplied by the event of it saying that we did not receive an email, as that would entail that there should have been an email.