A magician has $5$ coins. He initially places $3$ of the coins with heads-up and the rest with tails-up. Then he performs a process in which he flips one coin every second. The process stops when all the coins are tails-up. What is the probability that the process ends in $3$ seconds.

I found two methods two solve this but both are giving different answers.

Method 1: The process will end in exactly $3$ seconds when at each step the the coin with heads-up is flipped.

The probability of choosing one heads-up coin in first step is $\frac{3}{5}$.

Now we have flipped one heads-up coin. So, probability of choosing one heads-up coin in second step is $\frac {2}{5}$.

Similarly in third step it is $\frac{1}{5}$.

So, the probability that the process ends in three seconds is $(\frac{1}{5} \cdot \frac{2}{5} \cdot \frac{3}{5})=\frac{6}{125}$

Method 2: The process will end in $3$ seconds if we flip heads-up coin in all the steps.

We wi ll find all the possible sequences of steps:








Where $H$ or $T$ at the $i_{th}$ position represents the heads-up or tails-up coin respectively flipped at $i_{th}$ second.

So, the required probability is $\frac{1}{7}$

Why am I getting different answers? Which one is wrong?


Method $1$ is correct. Method $2$ is completely wrong: the probability of picking a heads-up coin to flip changes with every step.

  • $\begingroup$ Can you tell me what's wrong in second method? $\endgroup$ – Amit Shah Apr 4 '20 at 9:08
  • 1
    $\begingroup$ @AmitShah Just don't use it. I have given one reason already, but there are so many reasons it's wrong that I'd not be certain that I'd got them all. $\endgroup$ – Parcly Taxel Apr 4 '20 at 9:09
  • $\begingroup$ I couldn't understand your reason. $\endgroup$ – Amit Shah Apr 4 '20 at 9:11
  • 1
    $\begingroup$ @AmitShah There's also the fact that not all the sequences are equally likely. You can't just list all the sequences and say "oh, the probability of $HHH$ occurring is $\frac17$". It's like saying that the Large Hadron Collider will destroy the world with a 50% chance because either it will happen or it will not (and this claim was actually made to the press I think in 2008). $\endgroup$ – Parcly Taxel Apr 4 '20 at 9:14
  • 1
    $\begingroup$ @AmitShah Yes, a correct solution based on method $2$ would be far longer and excessive than one using method $1$. $\endgroup$ – Parcly Taxel Apr 4 '20 at 9:16

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.