Today's weather tomorrow's weather

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.