Answer:
a) Sunny Cloudy
T = [ 0.9 0.2 ] Sunny
[ 0.1 0.8 ] Cloudy
b) 0.72269
Step-by-step explanation:
Given that;
90% of all sunny days are followed by sunny days while 80% of cloudy days are followed by cloudy days;
i.e if today is sunny (90%), the next day is sunny so its 10% cloudy
and if today is cloudy (80%), the next day is cloudy so its 20% sunny
a)
Now let T be the transition probability matrix which is expressed as;
Sunny Cloudy
T = [ 0.9 0.2 ] Sunny
[ 0.1 0.8 ] Cloudy
b)
If today is sunny, to find the probability that it will be sunny in 5 days from now = P^5.x where x = [ 1 ] sunny
[ 0 ] cloudy
so
p^5X = P.P.P.P.P.x
P.x = [ 0.9 0.2 ] [ 1 ] = [ 0.9 ]
[ 0.1 0.8 ] [ 0 ] [ 0.1 ]
(P^2).x = [ 0.9 0.2 ] [ 0.9 ] = [ 0.83 ]
[ 0.1 0.8 ] [ 0.1 ] [ 0.17 ]
(P^3).x = p(p^2)x = [ 0.9 0.2 ] [ 0.83 ] = [ 0.781 ]
[ 0.1 0.8 ] [ 0.17 ] [ 0.219 ]
(P^4).x = p(p^3)x = [ 0.9 0.2 ] [ 0.781 ] = [ 0.7467 ]
[ 0.1 0.8 ] [ 0.219 ] [ 0.2533 ]
(P^5).x = p(p^4)x = [ 0.9 0.2 ] [ 0.7467 ] = [ 0.72269 ]
[ 0.1 0.8 ] [ 0.2533 ] [ 0.27731 ]
so (P^5).x = [ 0.72269 ] sunny
[ 0.27731 ] cloudy
therefore the probability that it is sunny 5 days from is 0.72269
