Download - Question 14 Exercise 16.02 page 341
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Question 14Exercise 16.02 page 341
Carwash
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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…
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On average 9 drivers per hour pay to use a carwash.
Mean = 9 per hour
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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.
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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.
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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
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Answer in the back of the book0.842813
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What went wrong?We ignored the cars already
waiting. (Thinking they would be INCLUDED in the Poisson
calculation).But they are EXTRA cars!
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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)
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Answer in the back of the book0.842813
(Note we were now CORRECT but did not know it!)
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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 15minsPLUS more than two in 5 mins!
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Lets look at the 15 minutes.What is the probability 2 or more
cars will arrive?Mean = 9 per hour2.25 per 15 minsp(x>1)=1-p(x<=1)
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So whilst the 3 are being washed there is a 0.65745 chance that 2 or
more will arrive!
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NOW Lets look at the last 5 minutes.What is the probability more than
one car will arrive?Mean = 9 per hourso 9/60 per minute
times by 5 to get per 5 minutes45/60 = 0.75
p(x>1)=1-p(x<=1)
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STOP: this is getting messy!
Continuing with this approachWe drew a probability tree.
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Which is the SAME answer as the p(x>2) – which was a lot quicker!
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Then we found a text book with the answer
0.80086hand written in the back of the book.
So WE WERE CORRECT!(a bit of rounding error)