What is the probability of observing 3 consecutive 0s in a b

What is the probability of observing 3 consecutive 0\'s in a bit string of length 7

What is the probability of observing 5 consecutive 1\'s in a bit string of length 7

What is the probability of observing either 3 consecutive 0\'s and 5 consecutive 1\'s in a bit string of length 7

Solution

For  3 consecutive 0\'s in a bit string of length 7:

Let An be the number of bit-strings of length n with 3 consecutive zeros. Then

An=2An1+2^n4An4

here A0 = A1=A2=0 A3=1 A4=3, A5=8 ,A6=20 ,A7=47

TOTAL cases = 2^7= 128

Probability = 47/128 = 0.3671875