medium | probability |

Snow-particles are falling on the ground one after another. A particular snowflake turns out to be of type "Stellar Dendrite" with probability 'p' if its previous particle was also Stellar Dendrite, and with probability 'q' if previous one was something else. If a snowflake is picked from ground, what is the probability that it is Stellar Dendrite?

Need to form a recursive equation of conditional probability

probability is q/(1-p+q)

Let $x$ be the probability that a snowflake picked from the ground is a 'Stellar Dendrite'. Thus, when a new snowflake is falling, the last snowflake was Stellar Dendrite with probability $x$.

This means that the probability the new falling snowflake is Stellar Dendrite $= x*p + (1-x)*q$. But, for the composition of the snowflakes on the ground to remain constant, $xp+(1-x)q$ should be same as $x$

$\implies x = xp + q - xq = x(p-q) + q$

$\implies x (1- (p-q)) = q$

$\implies x = \dfrac{q}{ (1- p + q)}$

This is a kind of steady state analysis.