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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:

$HHH$

$HHT$

$HTH$

$THH$

$HTT$

$TTH$

$THT$

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?

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

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  • $\begingroup$ Can you tell me what's wrong in second method? $\endgroup$ – Amit Shah Apr 4 '20 at 9:08
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    $\begingroup$ @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
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    $\begingroup$ @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
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    $\begingroup$ @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

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