Probability of Rain

It’s been a while, so let’s do a probability problem. I found this one in the youtube channel “MindYourDecisions“. If it rains on a day, the probability of rain tomorrow increases by 10%; if not, it reduces by 10% for the next day. What is the probability that it will rain forever within a few days if the chance to rain today is 60%?

Let P_x be the chance of raining forever, starting from a day x% rain. We can write the following equations. Note that P100 means it will rain today and every day from there. On the other hand, P0 suggest no rain today and, as a result, it continues.

P₁₀₀	1
P₉₀	0.9P₁₀₀ + 0.1P₈₀
P₈₀	0.8P₉₀ + 0.2P₇₀
P₇₀	0.7P₈₀ + 0.3P₆₀
P₆₀	0.6P₇₀ + 0.4P₅₀
P₅₀	0.5P₆₀ + 0.5P₄₀
P₄₀	0.4P₅₀ + 0.6P₃₀
P₃₀	0.3P₄₀ + 0.7P₂₀
P₂₀	0.2P₃₀ + 0.8P₁₀
P₁₀	0.1P₂₀ + 0.9_P0
P₀	0

Substituting the end value (P₀ = 0) for the term P₁₀ and working upwards,

P₁₀₀	1
P₉₀	0.9P₁₀₀ + 0.1P₈₀	0.998P₁₀₀
P₈₀	0.8P₉₀ + 0.2P₇₀	0.98P₉₀
P₇₀	0.7P₈₀ + 0.3P₆₀	0.93P₈₀
P₆₀	0.6P₇₀ + 0.4P₅₀	0.82P₇₀
P₅₀	0.5P₆₀ + 0.5P₄₀	0.67P₆₀
P₄₀	0.4P₅₀ + 0.6P₃₀	0.51P₅₀
P₃₀	0.3P₄₀ + 0.7P₂₀	0.35P₄₀
P₂₀	0.2P₃₀ + 0.8P₁₀	0.22P₃₀
P₁₀	0.1P₂₀ + 0.9P₀	0.1P₂₀
P₀	0

The value of the last term, 0.998P₁₀₀ can be evaluated by substituting P₁₀₀ = 1. Repeating the exercise, now downwards,

P₁₀₀	1
P₉₀	0.9P₁₀₀ + 0.1P₈₀	0.998P₁₀₀	0.998
P₈₀	0.8P90 + 0.2P70	0.98P90	0.98
P₇₀	0.7P80 + 0.3P60	0.93P80	0.91
P₆₀	0.6P70 + 0.4P50	0.82P70	0.75
P₅₀	0.5P60 + 0.5P40	0.67P60	0.55
P₄₀	0.4P50 + 0.6P30	0.51P50	0.34
P₃₀	0.3P40 + 0.7P20	0.35P40	0.18
P₂₀	0.2P30 + 0.8P10	0.22P30	0.08
P₁₀	0.1P20 + 0.9P0	0.1P20	0.02
P₀	0

Therefore, the required probability is 75%.

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