Question

On a given day the air quality in a certain city is either good or bad. Records show that when the air
quality is good on one day, then there is a 95% chance that it will be good on the next day, and when
the air quality is bad on one day, then there is a 45% chance that it will be bad on the next day.

Answer 1

(b) If the air quality is good today, what is the probability that it will be good two days from now?

Answer 2

i am not sure what i am doing wrong. question is related to markov chain rules

Answer 3

I can help. Can you show what you did so far?

Answer 4

i can type a bit... i have written in my rough copy so you cant understand
what i did is

Answer 5

.96G+0.55B=G eq 1
G+B=1 eq 2
i solve them by substitution

Answer 6

my answer is 0.9168 which should be 0.93.

Answer 7

well, I'd start with listing all the possible chains of events:

Answer 8

starting with good air (G), going to next day, and then going to day after that

Answer 9

i did by two method i use this foemula also (I-P)q=0

Answer 10

I'd see these possibilities: G->G->G and G->B->G

Answer 11

(we know we start with G and end with G)

Answer 12

i did that also. i got 0.91672

Answer 13

okay let's see, so the probability of the first GGG sequence is: 0.95*0.95

Answer 14

can you just show me the starting step how would you do like equations?

Answer 15

the prob of the other sequence GBG is 0.05*0.55

Answer 16

its corrct

Answer 17

sure i can show you the equations. Are you familiar with the conditional notation, like this P(G|G) ?

Answer 18

no

Answer 19

how we can solve this by markov chain rule?

Answer 20

P(G|G) stands for the prob that, given Good air quality today, we will see Good tomorrow

Answer 21

ok

Answer 22

The Markov chain rule says that by multiplying P(G|G)*P(G|G) we get P(GG|G), which is the probability that, given Good today, we will get Good tomorrow AND Good day after that.

Answer 23

so this allows us to just multiply the Markov probabilities together to get the probabilities of those sequences I was talking about

Answer 24

so for GGG, I multiply P(G|G)*P(G|G) and for GBG multiply P(G|B)*P(B|G)

Answer 25

those two sequences probabilities need to be added because they are both possible (one OR the other)

Answer 26

and so we get 0.93

Answer 27

can you make a tree diagram for me ? i got it now whats going on.

Answer 28

is that what you mean?