V 2 Rocket Hits Adapted from Richard Isaac

V 2 Rocket Hits Adapted from Richard Isaac, The Pleasures of Probability, Springer Verlag, 1995, pp. 99101. 576 0. 25 Km 2 areas of South London in a grid (24 by 24) 535 rockets were fired randomly into the grid = n P(a rocket hits a particular grid area) = 1/576 = 0. 001736 = θ Expected number of rocket hits in a particular area = 535/576 = 0. 92882 How many rockets will hit any particular area? 0, 1, 2, … could be anything up to 535. The 0. 9288 is the λ for the Poisson distribution:

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Poisson Process • • θ = 1/169 N = 133 λ = 133 * 1/169 = 0. 787 Probabilities: – – – P(X=0) =. 4552 P(X=1) =. 3582 P(X=2) =. 1410 P(X=3) =. 0370 P(X=4) =. 0073 P(X>4) =. 0013

Interpreting The Process • λ = 0. 787 • Probabilities: – – – P(X=0) =. 4552 P(X=1) =. 3582 P(X=2) =. 1410 P(X=3) =. 0370 P(X=4) =. 0073 P(X>4) =. 0013 • There are 169 squares • There are 133 “trials” • Expect. 4552*169 = 76. 6 to have 0 hits/square • Expect. 3582*169 = 60. 5 to have 1 hit/square • Etc. • Expect the average number of hits/square to =. 787.

Does the Theory Work? Theoretical Outcomes Sample Outcomes Outcome Probability Number Sample Proportion of Cells 0 . 4552 77 . 4733 80 1 . 3582 60. 5 . 2781 47 2 1410 23. 8 . 1420 24 3 0370 6. 3 . 0592 10 4 0073 1. 2 . 0118 2 >4 0013 0. 2 . 0000 0 n*λ =. 787 Number of cells 0(80)+1(47)+2(24)+. . . ]/169=. 787