Suppose only two weather conditions are considered, rain or no rain. If you know today's weather, the probability that tomorrow's weather will remain unchanged is p, and the probability that tomorrow'

Suppose only two weather conditions are considered, rain or no rain. If you know today's weather, the probability that tomorrow's weather will remain unchanged is p, and the probability that tomorrow's weather will become1-p. Hello:

Let Ai be: there is no rain on the first day, Pi=P(Ai), i= 1, 2. ..................................................................................................................................

P(A(i+ 1)|Ai)=P，P(A(i+ 1)|CuAi)= 1-P

By the total probability formula:

P (no rain on the nth day) =P (no rain on the nth day +0) *P (no rain on the nth day +0) +P (no rain on the nth day +0) *P (no rain on the nth day | 65438 +0)

Therefore, pn = PPN-1+(1-p) (1-PN-1) = (2p-1) pn-1-p Note: pn-.

The recurrence formula is: pn-1/2 = (2p-1) (pn-1-1/2).

So pn-1/2 = (2p-1) (n-1) (p1-/2).

Bring P 1= 1 into the above formula:

pn=( 1/2)*[ 1+(2p- 1)^(n- 1)]

If you are satisfied, please adopt it. Thank you.