An infinite collection of lily-pads are arranged at integer points a line, and a

Question

An infinite collection of lily-pads are arranged at integer points a line, and are indexed by {ellipsis, -2, -1, 0, 1, 2, ellipsis}. A frog is initially at lily-pad 0. At each time-step it hops one-step to either its left or right with equal probability. For example, suppose it is at lily-pad '-45'. Then at the next time-step, it can be at lily-pad '-44' or '-46'. After 10,000 time-steps, determine the expected distance that the frog has traveled from '0'. Further, using the central limit theorem, determine the probability that the frog is at-least a distance of 15 from '0' at time-step 10,000.

Explanation / Answer

1)The expected distance the frog would have travelled is 0 since it can move left or right with equal probaiblity

2) - Standard deviation = sqrt(4*n*p*(1-p))=sqrt(10000*4*0.5*0.5)=100

Z = (x-mean)/sigma = 15/100=0.15

P(Z>z) =P(z>0.15) = 0.440382 from normal distribution tables.

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