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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. Autograph Advanced Click Select Object Select Enter Probability Distributionthen select Binomial Enter the values of n and p Click Change the x values to Right click on the graph background Select Fit dependent Poisson 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. 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?……………………………………………………………………………………………… ………………………………………………………………………… Click Change Step to 5 Click 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?……………………………………………………………………………………………… ………………………………………………………………………………………………………… Repeat this for n=100 What is the difference in the values for P(x=16) for the Binomial and the Poisson?……………………………………………………………………………………………… ………………………………………………………………………… 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) Click 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 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