Answer :
Final answer:
The model of light in the room using a Markov chain has six states, representing the number of lit candles. Transitions between states occur at the change of each minute depending on whether a candle is lit or not. The transition probability matrix is created by calculating the probabilities of moving from one state to another.
Explanation:
In this problem, you have to model the level of light in a room using a Markov chain. There are six states: the room can have 0, 1, 2, 3, 4, or 5 candles lit. Each state represents the number of lit candles at a given time (after t minutes). The transition between states happens each minute when a single candle is chosen at random and either turned off (if it was lit) or turned on (if it was unlit).
To find the transition probability matrix, you need to calculate the probabilities of moving from one state to another. For example, the probability of going from 0 candles lit to 1 candle lit is 5/5, or 1, because there are 5 unlit candles. From 1 lit candle to 2, the probability is 4/5, because out of the five candles, there are 4 unlit and could be lit. The transition matrix for this problem is then as follows:
0: [0, 1, 0, 0, 0, 0]
1: [1/5, 0, 4/5, 0, 0, 0]
2: [0, 2/5, 0, 3/5, 0, 0]
3: [0, 0, 3/5, 0, 2/5, 0]
4: [0, 0, 0, 4/5, 0, 1/5]
5: [0, 0, 0, 0, 5/5, 0]
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The problem is discussing five candles in a room that has no other sources of light.
There are two states for each candle - lit or not lit. Each candle can either be lit or not lit. Every minute, one of the five candles is chosen at random, and its candle is put out or re-lit. If it was lit, it is turned not lit, and if it was not lit, it is lit. This model can be demonstrated as a Markov Chain with six states.
These states include 0 to 5, representing the number of lit candles in the room after t minutes. So, it has six states i.e., 0,1,2,3,4,5.
The probability transition matrix will be of size 6×6. Let P(i, j) be the probability of going from state i to state j. Then the probability of the candle that has been picked up will be turned on or off.
The new state will be reached. The probability of going to each state is calculated.
In the transition matrix, the probability of going from one state to another is recorded. Here's the probability transition matrix for each of the six states:0 → (0,1): 0.20, (1,0): 0.80;1 → (0,1): 0.20, (1,0): 0.20, (2,1): 0.60;2 → (1,2): 0.20, (2,1): 0.40, (3,2): 0.40;3 → (2,3): 0.20, (3,2): 0.60, (4,3): 0.20;4 → (3,4): 0.60, (4,3): 0.40;5 → (4,5): 1.0;Explanation:The transition probability matrix is calculated by finding the probability of moving from one state to another. So, in the given problem, we first find the states (0,1,2,3,4,5) and then, according to the rules, calculate the probability of going from one state to another.
The probability of the candle that has been picked up will be turned on or off, and the new state will be reached. For example, the transition probability from 0 to 1 is 0.20, which means that 20% of the time, one candle will be lit.
The transition probability from 1 to 2 is 0.60, which means that 60% of the time, two candles will be lit. And so on.
Summary: The given problem shows the calculation of the probability transition matrix for the level of light in a room, where five candles are placed, and no other source of light is available. A Markov Chain is developed with six states, where the number of lit candles in the room after t minutes is recorded. The transition probability matrix is calculated by finding the probability of moving from one state to another.
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