3
$\begingroup$

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?

$\endgroup$
2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Oh. Okk. I didn't realise that order mattered. $\endgroup$ – Tejas Nov 22 '14 at 15:00
  • 1
    $\begingroup$ 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. $\endgroup$ – David Nov 22 '14 at 15:01
  • $\begingroup$ That's right! Thanks. :) $\endgroup$ – Tejas Nov 22 '14 at 15:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.