Download Learning objective: To be able to determine when the Poisson

Comparing the Binomial and Poisson Distributions
Learning objective: To be able to determine when the Poisson Distribution can be used
as a good approximation to the Binomial Distribution.
An Archer fires 5 arrows at a target and for each arrow, independently of all the others, the
probability that it hits the bull’s eye is 0.125.
So we can use X  B(5, 0.125)
We can use Autograph to model this problem.
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Autograph
Advanced
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Click
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Select Object
Select Enter Probability Distributionthen select Binomial
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Enter the values of n and p
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Click
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Change the x values to
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Right click on the graph background
Select Fit dependent Poisson
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Click on the Binomial Distribution in the key at the bottom of the page.
This should highlight the edge of the coloured box in a thick black line.
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Right click on the background again
Select Table of Statistics
Min: -2
Max: 20
What is the difference in the values for P(x=2) for the Binomial and the
Poisson?………………………………………………………………………………………………
…………………………………………………………………………
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Click
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Change Step to 5
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Click
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Right click on the background again
Select Table of Statistics
until n = 60
S2 Ch 1 Binomial and Poisson Distributions
What is the difference in the values for P(x=12) for the Binomial and the
Poisson?………………………………………………………………………………………………
…………………………………………………………………………………………………………
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Repeat this for n=100
What is the difference in the values for P(x=16) for the Binomial and the
Poisson?………………………………………………………………………………………………
…………………………………………………………………………
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Repeat for other values of n
What do you notice?…………………………………………………………………
…………………………………………………………………………………………
Extension
Question 2 Ex1g p34 S2 book
An Archer fires arrows at a target and for each arrow, independently of all the others, the
probability that it hits the bull’s eye is 0.125.
a) Given that the archer fires 5 arrows, find the probability that fewer than 2 arrows hit
the bull’s eye
The archer now fires 60 arrows at the target. Using a suitable approximation find
b) The probability that fewer than 10 hit the bull’s eye
c) The smallest value of m such that the probability that the archer hits the bull’s eye is
greater than 0.5
Hints
a) and b)
To find P(x<r):
 Click on the distribution you wish to use in the key (This should highlight the black
line around that block)
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Click
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Select Cumulative and type in the value you need. Beware of the use of < and  in
the question and on the screen.
Click View then Status Bar to view the results of this calculation
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c)
Try moving the yellow diamond along the axis and see what changes this makes to
the status bar
S2 Ch 1 Binomial and Poisson Distributions