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雪花形态的马尔可夫链

Pattern on Snowflakes

专题
Probability / 概率
难度
L4

题目详情

雪花一片接一片落到地面。若上一片雪花是 Stellar Dendrite，则下一片成为 Stellar Dendrite 的概率为 pp；若上一片不是，则下一片成为 Stellar Dendrite 的概率为 qq。

从地面随机取一片雪花，问它是 Stellar Dendrite 的概率是多少？

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?

snowflake

Hint

Need to form a recursive equation of conditional probability

解析

答案是

q1−p+q.\frac{q}{1-p+q}.

设稳态下取到 Stellar Dendrite 的概率为 xx。则下一片为该类型的概率满足

x=xp+(1−x)q.x = xp + (1-x)q.

解得 x(1−p+q)=qx(1-p+q)=q，因此 x=q1−p+qx=\frac{q}{1-p+q}。

Original Explanation

probability is q/(1-p+q)

Solution

Let xx 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 xx.

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

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

  ⟹  x(1−(p−q))=q\implies x (1- (p-q)) = q

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

This is a kind of steady state analysis.