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?

PS:Although no two snowflakes are alike, yet there are various crystalline structures to categorize their interesting shapes. The image depicts the most popular shape, called Stellar Dendrites, which means star-like particles with tree-like branches.

PS:Although no two snowflakes are alike, yet there are various crystalline structures to categorize their interesting shapes. The image depicts the most popular shape, called Stellar Dendrites, which means star-like particles with tree-like branches.

Hint

Need to form a recursive equation of conditional probability

Answer

probability is q/(1-p+q)

Solution

Solution by Palak:

Let x be the probability that a snowflake picked from ground is Stellar Dendrite. Thus, when a new snowflake is falling, with prob=x the last snowflake was Stellar Dendrite => prob 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 =x => x=1/(1+(1-p)/q)

This is a kind of steady state analysis.

Let x be the probability that a snowflake picked from ground is Stellar Dendrite. Thus, when a new snowflake is falling, with prob=x the last snowflake was Stellar Dendrite => prob 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 =x => x=1/(1+(1-p)/q)

This is a kind of steady state analysis.

Source: Self

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