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?
Best Answer
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.