Question

REPOST: A biased coin has 0.75 probability of flipping heads and 0.25 of flipping tails. What is the probability that the sequence TTHH (T=tails, H=heads) only occurs after the 8th flip?

Answer 1

Given:-

P(Flipping Heads) = 0.75

P(Flipping Tails) = 0.25

We have to find the probability:

P(TTHH only occurs after the 8th flip) = 1 - P(TTHH occurs before the 8th flip)

Now

P(TTHH occurs before the 8th flip) = P(TTHHXXXX)+ P(XTTHHXXX) + P(XXTTHHXX) + P(XXXTTHHX)+ P(XXXXTTHH)

Here X can be either heads or tails

= (0.75)²*(0.25)²*(((0.75)⁰*(0.25)⁴*1+(0.75)¹*(0.25)³*4+(0.75)²*(0.25)²*6+(0.75)³*(0.25)¹*4+(0.75)⁴*(0.25)⁰*1)*5

= 0.17578125

Therefore

P(TTHH only occurs after the 8th flip) = 1 - P(TTHH occurs before the 8th flip) = 1 - 0.17578125 = 0.82421875