Consider a production system composed of two machines, where only one machine needs to be operational at any given time. The breakdown probability of any operational machine on any given day is 0.2. In case of a machine failure, the production is stopped for the day, and is resumed the next day with the other machine (if available). The repair job of the failed machine also starts the next day. It takes two days to repair a machine, and both machines can be repaired simultaneously. 1) Define the state space and draw the state transition diagram. 2) Starting with two available machines at the beginning of the first day, what is the probability that both machines are unavailable at the end of the 3rd day? 3) In the long run, what is the ratio of the days that both machines are unavailable?
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
Consider a production system composed of two machines, where only one machine needs to be operational at any given time. The breakdown probability of any operational machine on any given day is 0.2. In case of a machine failure, the production is stopped for the day, and is resumed the next day with the other machine (if available). The repair job of the failed machine also starts the next day. It takes two days to repair a machine, and both machines can be repaired simultaneously.
1) Define the state space and draw the state transition diagram.
2) Starting with two available machines at the beginning of the first day, what is the probability that both machines are unavailable at the end of the 3rd day?
3) In the long run, what is the ratio of the days that both machines are unavailable?
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