# Finding probability in the case of a biased coin

A coin is biased so that the probability of falling head when tossed is $\frac14$. If the coin is tossed $5$ times, find the probability of obtaining $2$ heads and $3$ tails, with heads occurring in succession.

I know that each toss is an independent event. And for independent events, $$P(A\cap B\cap C\cdots)=P(A).P(B).P(C)\cdots$$ Going by that, the answer to this question must be $$\frac14.\frac14.\frac34.\frac34.\frac34=\frac{3^3}{4^5}$$ However, the given answer is $$\frac{3^3}{4^4}$$

Where am I going wrong?

Good outcomes are HHTTT, THHTT, TTHHT, and TTTHH. You computed only the first one so your answer is only $1/4$th of the final answer. Order is important here making those $4$ results different.
• Think of it like this, based on your biased coin, what if someone asked you what is the probability of getting $2$ heads followed by $3$ tails. It would be your original answer. Then suppose they changed the question to your original question, the answer would increase to $4$ times that because the pair of adjacent heads can occur in $4$ different places in the $5$ coin tosses. – David Nov 22 '14 at 15:01