(1) If volume is high this week, then next week it will be high with a probability of 0.6 and low with a probability of 0.4. (ii) If volume is low this week then it will be high next week with a probability of 0.3. The manager estimates that the volume is five times as likely to be high as to be low this week.
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.
![**Understanding Probability Transitions in Markov Chains**
In this educational exercise, we explore a Markov chain model created to predict changes in volume levels over time. We begin by defining two possible states for volume:
- **State 1:** High Volume
- **State 2:** Low Volume
### Volume Transition Conditions
(i) If the volume is high this week, it has a 60% chance of staying high next week and a 40% chance of dropping to low.
(ii) If the volume is low this week, there’s a 30% probability it will become high next week.
### Manager's Estimate
The manager suggests that the volume is five times more likely to be high than low.
### Tasks
1. **Transition Matrix (P) Construction**
We establish a transition matrix P for the Markov chain, using the probabilities provided.
\[
P = \begin{bmatrix}
\_\_ & \_\_ \\
\_\_ & \_\_
\end{bmatrix}
\]
2. **State Vector \( X_0 \)**
Determine the initial state vector \( X_0 \) based on the manager's estimate of volume likelihood.
\[
X_0 = \begin{bmatrix}
\_\_ \\
\_\_
\end{bmatrix}
\]
3. **Future Predictions Using State Vector \( X_2 \)**
Using the estimate as the initial state vector, calculate the state vector for two weeks from now \( X_2 \).
\[
X_2 = \begin{bmatrix}
\_\_ \\
\_\_
\end{bmatrix}
\]
Finally, determine the probability that the volume will be high two weeks from now.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac73edbf-297f-4aed-9f9e-4c41e8d7aed8%2F2201274b-ef28-480d-819f-48d93dcf1808%2Fqflnudo_processed.png&w=3840&q=75)
![(4) Again, using the manager's estimate as the initial state vector, find the state vector for three weeks from now:
\[ X_3 = \left[ \begin{array}{c|c} \square & \square \end{array} \right] \]
What is the probability that three weeks from now the volume will be high?
\[ \square \]
(5) Suppose, contrary to the manager's estimate, that this week the volume is low. How many weeks must pass before a week comes along in which the probability of high volume is at least 0.3?
\[ \square \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac73edbf-297f-4aed-9f9e-4c41e8d7aed8%2F2201274b-ef28-480d-819f-48d93dcf1808%2F9qhen5_processed.png&w=3840&q=75)
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