python math small experiment (6) - how many times the coin toss to appear twice in a row up front?

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How many times the coin toss, two up front to appear?

The problem lies:
https://zhuanlan.zhihu.com/p/68358814

The core problem is the formula: $And (n_k | k-N- {1}) = e (k-N- {1}) + p + (p-1) (1 + E (n_k))$

Currently there have been continuous $k$ times face up, and only a thin last step. $E (n_k)$ First, at least equal to $And (N- {k-1})$
, Because I have been $k$ test times, this is the first that is right. Then the next step (I call it the key step) There are two cases: if it is face up, then it's done, because the probability is face-up
$p$ , so it corresponds to the second term of the right side of the equation; if it is negative up, then fall short, start all over again, which corresponds to the third term of the right side of the equation. Finally get the desired 6

Python simulation as follows:

import numpy as np 

def TossCoin(Consecutive_num):
    temp = 0    #记录连续次数
    count = 0   #记录掷硬币次数
    while temp < Consecutive_num+1:
        count += 1
        result = np.random.binomial(1,0.5)
        if result == 1:
            temp += 1
        else:
            temp = 0
        
        if temp == Consecutive_num:
            break  
    return count

TossCoin_result = [] 
for i in range(100000):
    TossCoin_result.append(TossCoin(2))

print(np.mean(TossCoin_result))

Output: 6.00694

experiment $p=1/6$ 3 times, as output: 259.2

According to the calculation $E(N_p)=\frac{1}{(1-p)}(\frac{1}{p^n}-1）$

Supplementary article: https://www.cnblogs.com/avril/archive/2013/06/28/3161669.html