Discrete and Continuous Distribution Hard Problem: Bees arrive at a flowerbed...

WillKissForHomework

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  1. Bees arrive at a flowerbed at a constant rate of 3 every 5 minutes. If an individual bee leaves and then comes back, it counts as another bee arrival. For the purposes of this problem, make the assumption that the arrival rate of the insects is the same every minute of every day you are observing them.
    1. Find the probability that less than 5 bees arrive in the next 10 minutes.
    2. Find the probability that at least 10 bees arrive in the next 15 minutes.
    3. Find the probability that it will be at least 3 minutes before the next bee arrives.
    4. Find the probability that the second bee arrives between 3 and 6 minutes from now. Express your answer as a function of the number e.
    5. Find the probability that the third bee arrives between 3 and 6 minutes from now. Express your answer as a function of the number e.
    6. It has been 5 minutes since the last bee arrived. What is the probability that the next bee arrived sometime in the next minute?
    7. When there are no bees in the flower bed, you leave for 5 minutes. When you come back, there are 5 bees there. What is the probability that at least 3 of them came in during the first 2 minutes that you were gone? (Assume that any bee that came is still there).
You count the number of bees that arrive in 5 minutes for consecutive 5 minute periods.


  1. Find the probability that fewer than 5 bees arrive in exactly 10 of the five-minute periods in the next hour.
  2. Find the probability that the sixth 5-minute period is the third one where the first bee arrived in less than 1 minute.
  3. You count the number of bees that come in each 5-minute period during an hour for 7 days in a row. What is the probability that on exactly one of these days, the 10th 5-minute period is the second one that had more than 4 bees arrive?

A biology class needs to count the number of bees that come to the flowerbed every 5 minutes, for 5 hours a day, for each of the next 10 days. They break into 5 teams, each of which is responsible for counting the bees for one hour each day.


  1. What is the probability that the 3rd team is the 1st one to see: at least 2 bees in each of their 5-minute periods on exactly 8 of the 10 days?
  2. What is the probability that the 4th team is the 1st one where the first bee arrives in less than 2 minutes in exactly 10 of their 5-minute periods, and this happens on exactly 1 of the 10 days?

Each team has 3 members, each of whom counts the bees for four 5-minute periods.


  1. Find the probability that at most 2 of the people on a team see less than 4 bees in at least one of their four 5-minute periods, one exactly one of the teams, on at exactly 2 of the 10 days.
  2. At the end of the 10 days, the total number of bees that came to the flowerbed during the 50 hours that it was observed is calculated. What is the probability that less than 1700 total bees visited?
  3. The flowerbed also attracts butterflies, which arrive at a rate of 3 every 10 minutes. What is the probability that there is a total of more than 20 bees and butterflies (added together) in the next 20 minutes? Justify why you can do what you did!
 
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