Use randomized simulation to estimate the expected length of the longest run of consecutive heads in a series of n coin flips. For example, in this series of 10 flips:

T, T, H, T, H, T, H, H, H, T 

the length of the longest run of heads is 3. Start with code in coins.py. First, modify doNCoinFlips function to return (instead of the numbers of heads and tails) the length of the longest run of consecutive heads found in a sequence of n flips. Next, modify doCoinFlipTrials to collect and process the results of multiple trials of coin flipping, calculating the average longest run of heads for the set of trials. Third, modify doCoinFlipExperiment to calculate and print the expected longest heads run length for a wide range of numbers of flipped coins.

coins.py:

import random import math def stdDev(values): mean = float(sum(values))/len(values) diffsum = 0.0 for item in values: diff = item - mean diffsum = diffsum + diff*diff return math.sqrt(diffsum/len(values)) # simulate flipping a coin numFlips times. # return tuple (number of heads, number of tails) # def doNCoinFlips(numFlips): numHeads = 0 flipNum = 0 while flipNum < numFlips: value = random.randint(1,2) if value == 1: numHeads = numHeads + 1 flipNum = flipNum + 1 return (numHeads, numFlips-numHeads) # Do numTrials coin flip simulations with numCoins each time. # For each trial compute ratio of number of heads to number of trials. # Compute average and standard deviation of the ratios over all trials. # Print summary and # return tuple (average heads/tails ratio, standard devation of ratio) # def doCoinFlipTrials(numTrials, numCoins): headTailRatios = [] headTailDiffs = [] for trial in range(numTrials): nHeads, nTails = doNCoinFlips(numCoins) #print("Trial #{}: numHeads = {}, numTails = {}".format(trial, nHeads, nTails)) # NOTE: This code *can* crash - it's unlikely but nTails can, of course, be 0 headTailRatios.append(float(nHeads)/nTails) ratiosAvg = sum(headTailRatios)/len(headTailRatios) ratiosStdDev = stdDev(headTailRatios) print("# of flips = {}: avg head/tail ratio: {} (SD: {})".format(numCoins, ratiosAvg, ratiosStdDev)) return (numCoins, ratiosAvg, ratiosStdDev) # Execute doCoinFlipTrials for different numbers of coins (and given number of trials), # starting with minCoins and increasing by factor until number exceeds maxCoins. # Return list of results returned by the calls to doCoinFlipTrials. # Thus, results will be of form [(minCoins, ..ratio.., ...std...), (factor*minCoins, ..ratio.., ...std..), ...] # def doCoinFlipExperiment(minCoins, maxCoins, factor, numTrials=20): numCoins = minCoins results = [] while numCoins <= maxCoins: results.append(doCoinFlipTrials(numTrials, numCoins )) numCoins = numCoins * factor return results # Uncomment section below if you have pylab ''' import pylab # results = doCoinFlipExperiment(16, 2**20, 2) # plotResults(results) # pylab.show() # # # Create figure with two subplots, one showing avg. head/tail ratios for different numbers of # coin flips, the other showing standard deviation of head/tail ratios. # def plotResults(results): numFlips, ratios, ratioSDs = tuple(zip(*results)) pylab.figure(1) pylab.clf() pylab.subplot(121) pylab.title("Head/tail ratios") pylab.xlabel("Number of flips") pylab.ylabel("Average head/tail ratio") pylab.semilogx() pylab.plot(numFlips, ratios, 'ro-') pylab.subplot(122) pylab.title("Std dev. of h/t ratios") pylab.xlabel("Number of flips") pylab.ylabel("Standard deviation") pylab.semilogx() pylab.plot(numFlips, ratioSDs, 'ro-') '''