Relatively Simple Probability Question

[gajohnson]

Homework Statement

This problem introduces a simple meteorological model, more complicated
versions of which have been proposed in the meteorological literature. Consider
a sequence of days and let Ri denote the event that it rains on day i. Suppose
that P(R | R ) = α and P(Rc | Rc ) = β. Suppose further that only today’s i i−1 i i−1
weather is relevant to predicting tomorrow’s; that is, P ( Ri | Ri −1 ∩ Ri −2 ∩ · · · ∩ R0) = P(Ri | Ri−1).

a. If the probability of rain today is p, what is the probability of rain tomorrow?
b. What is the probability of rain the day after tomorrow?
c. What is the probability of rain n days from now? What happens as n approaches infinity?

Homework Equations

NA

The Attempt at a Solution

Parts a and b are easy enough--just simple applications of the Law of Total Probability. However, I am having trouble with part c. Any help would be greatly appreciated.

It does appear that this question was previously asked, and someone gave this idea: p(⋂i=1nRi), but I'm not sure how to see this.

Thank you!

 

[Ray Vickson]

Homework Statement

This problem introduces a simple meteorological model, more complicated
versions of which have been proposed in the meteorological literature. Consider
a sequence of days and let Ri denote the event that it rains on day i. Suppose
that P(R | R ) = α and P(Rc | Rc ) = β. Suppose further that only today’s i i−1 i i−1
weather is relevant to predicting tomorrow’s; that is, P ( Ri | Ri −1 ∩ Ri −2 ∩ · · · ∩ R0) = P(Ri | Ri−1).

a. If the probability of rain today is p, what is the probability of rain tomorrow?
b. What is the probability of rain the day after tomorrow?
c. What is the probability of rain n days from now? What happens as n approaches infinity?

Homework Equations

NA

The Attempt at a Solution

Parts a and b are easy enough--just simple applications of the Law of Total Probability. However, I am having trouble with part c. Any help would be greatly appreciated.

It does appear that this question was previously asked, and someone gave this idea: p(⋂i=1nRi), but I'm not sure how to see this.

Thank you!

You need parentheses, or else use subscripts; that is, write either R_(i-1) or R_{i-1} or R_i-1 (obtained by using the X₂ button on the menu atop the input panel).

Anyway, your model is apparently a 2-state Markov chain with states 1 = R (rain) and 2 = N (no rain), and with 1-step transition probability matrix P = (p_{ij}), where p_{ij} = P(i-->j in one step), for i = 1,2 and j = 1,2. Also, you are given p_{11} = α and p_{22} = β. You can then get p_{12} and p_{21} (how?).

For part (c) you need to know some properties of Markov chains. I don't know the course context of the questions you are asking--for example, whether or not your course notes and/or textbook treat Markov chains)--so I don't know whether any advice I offer is irrelevant, but you can look for free on-line material, such as:
http://en.wikipedia.org/wiki/Examples_of_Markov_chains
or
http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter11.pdf (the first few pages)
or
http://www.math.ucla.edu/~pejman/intro2prob/LiveMeeting10.pdf .

There are also some other good sites with examples, etc., but some of them use the somewhat non-standard convention of having the transition-matrix *columns* sum to 1, instead of the more common and convention of *rows* summing to 1. In other words, their "transition matrices" are transposes of the usual transition matrices.

 

[haruspex]

Parts a and b are easy enough

To make sure we're all on the same page when attacking (c), pls post your answers for (a) and (b).

 

[gajohnson]

To make sure we're all on the same page when attacking (c), pls post your answers for (a) and (b).

Good idea.

For part a, I've got

pα+(1-p)(1-β)

And for part b, I've got

(pα+(1-p)(1-β))α + (1-pα-(1-p)(1-β))(1-β)

Both of these were found by simply using the Law of Total Probability. Are these correct? I tried extending this out to the third day to look for a pattern, but I didn't see anything. We have not yet covered Markov Chains, so I don't know if that is the route I ought to use for part c. Thanks again for your help!

 

[haruspex]

Yes, those are correct for a and b.

I tried extending this out to the third day to look for a pattern, but I didn't see anything. We have not yet covered Markov Chains, so I don't know if that is the route I ought to use for part c.

You could let r_n be the probability of rain on the nth day and obtain a recurrence relation for the sequence.

 

[gajohnson]

Yes, those are correct for a and b.

You could let r_n be the probability of rain on the nth day and obtain a recurrence relation for the sequence.

Thanks for your answer! Unfortunately I am not familiar with recurrence relations. Any intuition as to where to start, or is there another way to go about it that you see?

 

[haruspex]

With Markov chains and recurrence relations out it's getting a bit tough.
Can you make an inspired guess as to the shape of the answer and use induction to prove it? Start with last part - what do you think the limit will be?

 

[gajohnson]

With Markov chains and recurrence relations out it's getting a bit tough.
Can you make an inspired guess as to the shape of the answer and use induction to prove it? Start with last part - what do you think the limit will be?

I'm guessing it's not sufficient to write that the probability of rain n days from now is simply

P(R_n) = (P(R_n-1))α + (1-R_n-1)(1-β)

My intuition might be that the limit is α(1-β), but I can't support that very well. Is any of this in the ballpark? Thanks for bearing with me here.

 

[Ray Vickson]

Good idea.

For part a, I've got

pα+(1-p)(1-β)

And for part b, I've got

(pα+(1-p)(1-β))α + (1-pα-(1-p)(1-β))(1-β)

Both of these were found by simply using the Law of Total Probability. Are these correct? I tried extending this out to the third day to look for a pattern, but I didn't see anything. We have not yet covered Markov Chains, so I don't know if that is the route I ought to use for part c. Thanks again for your help!

Both of the above are OK.

Since you don't know Markov chains yet, it is difficult to do a good job on (c), but here is a little hint for an approach that seems intuitive (and is ultimately justifiable). Suppose the probability p_n of (Rn) rain on day n approaches a limit v for n → ∞. If this is so, we must, in the limit, get the same value v for p_{n+1}.