1. Question 14 Exercise 16.02 page 341 Carwash

2. This records our frustration with trying to match our answer with the back of the book. Learning did happen along the way! I uploaded it to show how tricky these problems can be…….. Personally I found the wording in this hard to understand…

3. On average 9 drivers per hour pay to use a carwash. Mean = 9 per hour

4. Each car-wash takes 5 minutes. The carwash closes at 7pm. A car leaves the carwash at 6:40 pm, when there are three cars in line.

5. a) Assuming a Poisson distribution is an appropriate model for the number of drivers per hour that pay for a car-wash, calculate the probability that there will be one or more drivers waiting in line at closing time.

6. The mean for the 20 minutes is 3. The three cars in line will be finished by 6:55pm allowing for one more car to arrive and be washed. So we are looking at p(x>4) p(x>4)=1-p(x<=3) = 1 – 0.64723188 = 0.3528

8. Answer in the back of the book 0.842813

9. What went wrong? We ignored the cars already waiting. (Thinking they would be INCLUDED in the Poisson calculation). But they are EXTRA cars!

10. Lets go back and re-read the question. Perhaps we ignore the ones in line and find the probability that more than one car will come in the 20 minutes. (6:40 to 7pm = 20 mins = 4 cars) P(x>1)= 1 – p(x<=1)

12. Answer in the back of the book 0.842813 (Note we were now CORRECT but did not know it!)

13. Ok – wrong AGAIN!!! PERHAPS we need to split the time – 15min and the last 5 mins. More than one in 15 mins AND more than one in 5 PLUS more than two in 15mins PLUS more than two in 5 mins!

14. Lets look at the 15 minutes. What is the probability 2 or more cars will arrive? Mean = 9 per hour 2.25 per 15 mins p(x>1)=1-p(x<=1)

16. So whilst the 3 are being washed there is a 0.65745 chance that 2 or more will arrive!

17. NOW Lets look at the last 5 minutes. What is the probability more than one car will arrive? Mean = 9 per hour so 9/60 per minute times by 5 to get per 5 minutes 45/60 = 0.75 p(x>1)=1-p(x<=1)

19. STOP: this is getting messy! Continuing with this approach We drew a probability tree.

20. Which is the SAME answer as the p(x>2) – which was a lot quicker!

21. Then we found a text book with the answer 0.80086 hand written in the back of the book. So WE WERE CORRECT! (a bit of rounding error)