Sunny skies over Titan?

[bonanova]

The findings of the Huygens probe indicate that Titan (Saturn's largest moon) has a nitrogenous atmosphere that periodically produces rain onto that moon's surface. Titan and Earth are the only known heavenly bodies with liquid rain. But given its hostile temperature of -180^oC Titan's rain is not water, it's liquid methane. But enough of the cold facts.

The exciting part of this puzzle is that in 2004 you were given a large supply of beef jerky, a warm parka, and the job of being Titan's chief in-person, feet on the ground, up close and personal, weather observer. Upon your recent return, you reported your findings on Titan's rain activity. Let's call the days on Titan that it did not rain "sunny" days, even with the constant nitrogenous smog. (You grew up in Los Angeles.) You found that sunny days on Titan were followed by another sunny Titan day 29 days out of 30, (p_ss = 29/30), while rainy days were followed by another rainy day with probability p_rr = 0.7.

Like many heavenly satellites, Titan's rotation is tidally locked to its orbital period (16 Earth days.) If you were on Titan for say 9 Earth years, on about how many Titan days did you observe rain?

[bushindo]

The findings of the Huygens probe indicate that Titan (Saturn's largest moon) has a nitrogenous atmosphere that periodically produces rain onto that moon's surface. Titan and Earth are the only known heavenly bodies with liquid rain. But given its hostile temperature of -180^oC Titan's rain is not water, it's liquid methane. But enough of the cold facts.

The exciting part of this puzzle is that in 2004 you were given a large supply of beef jerky, a warm parka, and the job of being Titan's chief in-person, feet on the ground, up close and personal, weather observer. Upon your recent return, you reported your findings on Titan's rain activity. Let's call the days on Titan that it did not rain "sunny" days, even with the constant nitrogenous smog. (You grew up in Los Angeles.) You found that sunny days on Titan were followed by another sunny Titan day 29 days out of 30, (p_ss = 29/30), while rainy days were followed by another rainy day with probability p_rr = 0.7.

Like many heavenly satellites, Titan's rotation is tidally locked to its orbital period (16 Earth days.) If you were on Titan for say 9 Earth years, on about how many Titan days did you observe rain?

Here's my answer

So, given that we start with a clear day, it would take an expected 1/( 1/30) = 30 days to get a rainy day.

Once we have a rainy day, it would take an average of (1/.3) = 3.333333… days until we get a clear day again.

So, on average we would have (3 + 1/3) rainy days for every (33 + 1/3) days, which means that the weather on Titan is sunny 90% of the time.

[bonanova]

Sounds almost intuitive when put that way, and no arcane expressions. Nice solve.

Never mind counting the days, a percentage is fine.

I hope the residual methane fumes clear up soon!