00:01
For this exercise, we refer back to exercise 16, which was the markov chain, which had to do with how many umbrellas sarah had at home.
00:10
And in exercise 16, we had come up with a single -step transition matrix here.
00:17
And now we are asked in part a that if sarah has two umbrellas at home on sunday night, what is the probability of having exactly two umbrellas at home on the next friday night? so that is exactly five days later, five working days later.
00:35
So we can say n is equal to five.
00:39
And so we are looking for the five -step transition probability of going from two to two.
00:50
And as usual, the way to find the transition probabilities for five steps is to take the single -step transition matrix and raise it to the exponent five.
01:01
So you can do that in software.
01:03
And so then this is equal to the entry 2 -2 in that matrix.
01:18
Now we do have to be careful because when we came up with this transition matrix in exercise 16, i had indexed the rows and columns as follows, just so it was easier to keep track of how many umbrellas were at home.
01:38
So here when i say entry 2 -2, i actually mean this entry right here, which is entry 3 in conventional matrix terminology.
01:53
And in the five -step transition matrix, that number happens to be 0 .274.
02:02
So that's sort of the first question of part a, and the second question is, what is the probability by friday evening that there will be at least two umbrellas at the house? so this time we're looking for the five -step probability, going from 2 to 2, plus the 5 -step probability.
02:26
Of going from 2 to 3, as well as from 2 to 4.
02:40
And so this is equal to 0 .274 plus 0 .279 plus 0 .279, plus 0 .221, which comes out to 0 .775.
03:16
So now for part b, we are given that there are two umbrellas on sunday night at home, and what are the chances that there will be zero umbrellas to take to work on the following? thursday morning.
03:30
So that is the same as asking the question, what is the probability that there will be zero umbrellas at home on wednesday night after work? so that is three steps from sunday night to wednesday night.
03:45
So that means we're looking for the three -step probability of going from two umbrellas at home to no umbrellas at home.
03:56
So this time, of course, we calculate, we calculate p3, which is the three -step transition matrix for the markov chain.
04:09
And then we look at entry 3 -1, which corresponds to going from 2 to 0.
04:22
Just to make that clear, we'd be going for entry 3 -1, which would be this entry, except that this is the single -step transition matrix.
04:33
So that would be the probability of going from 2 to 0 umbrellas.
04:38
But of course, these are actually row 3, and column 1.
05:02
And this comes out to 0 .038.
05:12
And last for part c, we're told that sarah has two umbrellas at home at the start of the week, and we're asked what the expected number of umbrellas she'll have at home at the end of monday and at the end of tuesday is.
05:27
So when we look at a transition matrix, so row 3, which corresponds to having two umbrellas, it is the probability distribution for how many umbrellas are going to be at home after the next transition...