The weather on any given day in a particular city can be sunny, cloudy, or rainy. It has been observed to be predictable largely on the basis of the weather on the previous day. Specfically:
if it is sunny on one day, it will be sunny the next day 1/2 of the time, and never be cloudy the next day
if it is cloudy on one day, it will be sunny the next day 1/3 of the time, and never be cloudy the next day
if it is rainy on one day, it will be sunny the next day 2/3 of the time, and be cloudy the next day 1/6 of the time
Using 'sunny', 'cloudy', and 'rainy' (in that order) as the states in a system, set up the transition matrix for a Markov chain to describe this system.
Use your matrix to determine the probability that it will rain on Wednesday if it is sunny on Sunday.
find matrix P and probability of rain on wednesday
To set up the transition matrix, we'll arrange the probabilities in a matrix with the rows representing the current day's weather and columns representing the next day's weather. The given information will help us fill probablilities for the transition matrix.
Here's the transition matrix P:
```
| Sunny | Cloudy | Rainy
-------+-------+--------+-------
Sunny | 1/2 | 0 | 1/2
Cloudy | 1/3 | 0 | 2/3
Rainy | 2/3 | 1/6 | 1/6
```
P = [[1/2, 0, 1/2],
[1/3, 0, 2/3],
[2/3, 1/6, 1/6]]
For the probability of rain on Wednesday given it was sunny on Sunday, we'll need to find the state probabilities after 3 days (Sunday to Wednesday). To do this, we raise the transition matrix P to the power of 3:
P^3 = P x P x P
Computing P^3, we get:
P^3 = [[9/16, 1/16, 6/16],
[7/27, 2/27, 18/27],
[49/108, 7/108, 52/108]]
Now, let's use the initial state where it was sunny on Sunday:
The initial state vector is: [1, 0, 0]
Multiply the initial state vector by P^3 to get the state probabilities on Wednesday:
State probabilities = [1, 0, 0] x P^3 = [9/16, 1/16, 6/16]
The probability of rain on Wednesday is 6/16, or 0.375.
To set up the transition matrix for this Markov chain, we need to define the probabilities of transitioning from one state to another. Let's denote the states as follows: S for sunny, C for cloudy, and R for rainy.
The transition probabilities can be summarized as follows:
- If it is sunny on one day, there is a 1/2 probability of it being sunny the next day and a 0 probability of it being cloudy or rainy.
- If it is cloudy on one day, there is a 1/3 probability of it being sunny the next day and a 0 probability of it being cloudy the following day.
- If it is rainy on one day, there is a 2/3 probability of it being sunny the next day and a 1/6 probability of it being cloudy the following day.
Now, we can set up the transition matrix P:
S C R
| 1/2 | 0 | 0 |
P = | 1/3 | 0 | 0 |
| 2/3 | 1/6 | 0 |
This is a 3x3 matrix, where the (i,j) entry represents the probability of transitioning from state i to state j.
To find the probability that it will rain on Wednesday if it is sunny on Sunday, we need to multiply the initial state vector by the transition matrix raised to the power of the number of days between Sunday and Wednesday (which is 3 in this case), and then extract the probability for the rainy state.
Let's denote the initial state vector as V0:
V0 = [1, 0, 0]
We can calculate the probability as follows:
V3 = V0 * P^3
Performing the matrix multiplication:
V3 = [1, 0, 0] *
| 1/2 | 0 | 0 |^3
| 1/3 | 0 | 0 |
| 2/3 | 1/6 | 0 |
V3 = [1, 0, 0] *
| 1/8 | 0 | 0 |
| 1/27 | 0 | 0 |
| 7/54 | 1/36 | 0 |
V3 = [1/8, 0, 0] + [1/27, 0, 0] + [7/54, 1/36, 0]
= [17/216, 1/36, 0]
Therefore, the probability of rain on Wednesday, given that it is sunny on Sunday, is 0.
To set up the transition matrix for this Markov chain, we need to organize the given probabilities into the matrix format.
Let's define the three states as follows:
- State 1: Sunny
- State 2: Cloudy
- State 3: Rainy
Now, let's set up the transition matrix P, where each element P(i, j) represents the probability of transitioning from State i to State j.
Since the question states the probability of transitioning from one state to another for each given state, we can fill in the transition matrix accordingly.
P =
| 1/2 1/3 2/3 |
| 1/2 0 1/6 |
| 0 2/3 1/6 |
This is the transition matrix for the given system, where each row represents the current state, and each column represents the next state.
To find the probability of rain on Wednesday given that it is sunny on Sunday, we need to calculate the probability of transitioning from sunny (State 1) to rainy (State 3) in two steps (Sunday to Monday, then Monday to Wednesday).
Let's denote:
- State 1: Sunny (S)
- State 2: Cloudy (C)
- State 3: Rainy (R)
To find the probability, we multiply the probability of transitioning from sunny to any other state on Monday (S to C and S to R), and then from that state to rainy on Wednesday.
P(rain on Wednesday | sunny on Sunday) = P(Sunday to Monday) * P(Monday to Wednesday)
P(Sunday to Monday) = P(S, S) = 1/2 (Sunny to Sunny)
P(Monday to Wednesday) = P(S to R) = 2/3 (Sunny to Rainy)
Thus,
P(rain on Wednesday | sunny on Sunday) = (1/2) * (2/3) = 1/3.
Therefore, the probability that it will rain on Wednesday, given that it is sunny on Sunday, is 1/3.