Markov Chains Rainy Vacation assume that if today is rainyth

Markov Chains Rainy Vacation: assume that if today is rainy,there is a 75% chance that tomorrow will be rainy, and likewise if today is sunny, there is a 75% chance that tomorrow will be sunny. The day you arrived on your vacation was sunny. What is probability of 3 consecutive rainy days in the week.

i found transition matrix with initial (1,0) sunny rain sunny 0.75 0.25 rain 0.25 0.75

I understand, if i multiply my matrix 6 times by itself and multiply by initial i get probabilities for rain day at day 7. But How i can use transition matrx to find probability of 3 consecutive rainy days in the week. Analytically, i undrstand

for 3 days S-R-R-R-S-S-S** S-S-R-R-R-S-S** S-S-S-R-R-R-S** S-S-S-S-R-R-R**

for 4 days S-R-R-R-R-S-S** S-S-R-R-R-R-S** S-S-S-R-R-R-R**

for 5 days S-R-R-R-R-R-S** S-S-R-R-R-R-R**

for 6 days S-R-R-R-R-R-R

My question: HOW I CAN USE MY TRANSITION MATRIX TO FIND PROBABILITY of 3 consecutive rainy days in the week.?

Solution

Transition matrix is helpful to calculate the probability of the future events. As we know that markovian properties say that future events depands on prsent valuue not the past value. In the transition matrix having elements is probability at different stage ( day).