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I was solving a probability problem which is stated like that:

Consider there are three seats and three people. First person seats on a random seat and two others choose their seats based on the following logic: if their seat is free they take it, if it's taken they take a random seat. What is the probability that the last person will take their seat?

The solution I came up with is the following:

$$A\ -\ last\ person\ seats\ in\ their\ seat$$ $$B_i\ -\ first\ person\ seats\ in\ the\ seat\ i$$ $$P(A)=P(B_1)\cdot{P(A|B_1)} + P(B_2)\cdot{P(A|B_2)} + P(B_3)\cdot{P(A|B_3)}={1\over3}\cdot1+{1\over3}\cdot{1\over2}+{1\over3}\cdot0={1\over2}$$

This answer is correct according to the book. What I dont really undetrstand is why I can't calculate it like that: $$P(A)=P(B)\cdot{P(A|B)} + P(1-B)\cdot{P(A|1-B)}={1\over3}\cdot1+{2\over3}\cdot{1\over3}={5\over9}$$ where B is event of the first passenger landing in their seat. My logic is that when first passenger chooses first seat the last one will always land in their seat so conditional probability here is 1. If the first passenger doesn't choose their seat we have 3 cases(2,1,3|3,1,2|3,2,1) from which only one suffices us, so conditional probability is $${1\over3}$$ here. $$B\ and\ 1-B$$ are exclusive so it's not the problem here obviously. I feel that I'm missing something very trivial here.

s0nicYouth
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    This is the Airplane Probability problem – lulu Feb 01 '23 at 13:09
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    To your specific question, if the first passenger does not take their own seat then the only way the third passenger does get their own seat is if the first passenger takes the second seat (probability $\frac 12$, given they didn't take their own) AND the second passenger then takes the first seat (probability $\frac 12$). thus the conditional probability is $\frac 14$ so you have $\frac 13\times 1 +\frac 23\times \frac 14=\frac 12$ as desired. – lulu Feb 01 '23 at 13:14
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    @lulu thanks for pointing out to the problem. Regarding your answer - basically I can't take 1/3 as conditional probability because the probability of each of those 3 outcomes are not equal(1/4,1/4, 2/4). I think I got it. Thank you for your answer. – s0nicYouth Feb 01 '23 at 14:06

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