Question

PLEASE ANSWER FROM WHERE IT SAYS "Continuing from Part 1"

Consider a sequence of n Bernoulli trials with success probabilty p per trial. A string of consecutive successes is known as a success run. Write a function that returns the counts for runs of length k for each k observed in a dictionary. For example: if the trials were [0, 1, 0, 1, 1, 0, 0, 0, 0, 1), the function should return {1:2, 2: 1)) What is the return value for the input np.random.randint(0,2,1000000)? Continuing from Part 1, what is the probability of observing at least one run of length 5 or more when n=100 and p=0.5?. Estimate this from 100,000 simulated experiments. Is this more, less or equally likely than finding runs of length 7 or more when p=0.7? Hint: you can use numpy's random choice function to generate the simulated experiments. For instance, np.random.choice([0,1], (expts, n), p= (1-2, p)) generates 'expts' many experiments as n dimensional arrays, where each component of the array is success (i.e. 1) with probability p.

Answer 1

I ran the length 5 and p = 0.5 eight times on my PC. The probability came around 0.53

I ran the length 7 and p = 0.7 five times on my PC. The probability came around 0.507

So scenario 1 definitely has greater probability.

Here's the code if you want to try it out. The code will take a few seconds to compute, especially if its an older PC.

import numpy as np
from collections import Counter
def count(ls,n): #n here defines the length of runs
    ls = ''.join(ls)
    #ms = ls.split('0')
    #ms = [i for i in ms if i]
    #cnt = Counter(ms)
    #print(cnt)
    srch = '0' + ('1'*n) + '0'
    if srch in ls:
        return 1
    else : return 0
expts = 100000
n = 100
run_length = 5 #change length of successive 1's (runs)
pr = 0.5 #change probability
ls = np.random.choice([0,1],(expts,n),p =(1-pr,pr))
fav = 0
for i in ls:
    i = list(map(str,i))
    a = count(i,run_length)
    fav = fav + a
    #print(a)
print(float(fav/expts))
#ls = [0,1,0,1,1,0,0,0,0,0,0,0,0,0,0]

#count(ls1)

Let me know if you want help figuring out how to change the parameters, I've commented what you'll need though.

Oh, and the commented counter lines are if you want to see how it runs, just remember to set the expts to around 100-1000 if you want to try to run them.