One alien lands on a planet. Every day, the alien can perform the following actions, with their probabilities as:
Self - destruct : 1/4 chance
Do nothing - 1/4 chance
Split into three aliens - 1/4 chance
Split into two aliens - 1/4 chance
If the alien splits, each produced alien has the same probabilities. What is the probability of the species going extinct?


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Take the probability of an alien and it's entire line dying out be D. So we can write D as

$D$ = $\frac14$(if it self destructs) +$\frac14$(if it does nothing)$\cdot$P(probability of going extinct tomorrow) + $\frac14$(if it splits into $3$)$\cdot$P(probability of all three going extinct) +$\frac14$(if it splits into $2$)$\cdot$P(probability of all two going extinct)

We can see that

P(probability of going extinct tomorrow) is same as today = $D$

P(probability of all three going extinct) = $D\cdot D\cdot D$ = $D^3$ as each of them has to go extinct which means it's an intersection of each dying case, which in case of independent events means multiplication.

Similarly, P(probability of all two going extinct) = $D\cdot D$ = $D^2$

Putting them in the equation gives us

$$D = \frac{(1+D+D^2+D^3)}{4}$$

$$D^3+D^2-3D +1 =0$$

This has $1$ as an obvious solutions, which helps us determine the other two $\sqrt{2}-1$ and $-\sqrt{2}-1$. Since probability cannot be negative, the latter is out of the question. Now between $1$ and $\sqrt{2}-1$, you can use either Expectation to see that number of aliens the next day will be $1.5>1$ which means there is non zero chance it might survive or you can use induction to get the fact that $P_k$(probability it goes extinct after $k$ days) is always less than $\sqrt{2}-1$, starting from $P_1$ which is $0.25$

This means

$$D = \sqrt{2}-1$$


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