Rain or Shine

Categorise the weather as either "wet" or "dry". The probability that tomorrow is "wet" depends upon the weather for some previous time. We use the notation : p(tomorrow is wet | today is wet) for the conditional probability that tomorrow is wet given that today is wet.

(i) write down p(tomorrow is dry | today is wet)
and p(tomorrow is dry | today is dry)

On 1^st June, when you are planning a picnic, the weather will be either wet or dry. Let p(June 1st is wet) = p and consequently p(June 1st is dry) = 1 - p

(ii) Write down p(June 2nd is wet) in terms of p.

Let p _n denote the probability that the weather on day n is wet given that the weather the day before is wet. Now simplify the problem by making the assumption that the weather on any given day depends only on the weather the day before and not on the weather before then.

Use an iterative technique (with your calculator if you wish) to find

(iii) p(July 1 ^st is wet | June 1 ^st is wet)

(iv) p(July 1 ^st is wet | June 1 ^st is dry)

(v) What do the answers to (iii) and (iv) tell you about long-term future weather?