Math Problem Statement
At one liberal arts college, students are classified as humanities majors, science majors, or undecided. The chances are 4040% that a humanities major will change to a science major from one year to the next, and 2525% that a humanities major will change to undecided. A science major will change to humanities with probability 0.350.35, and to undecided with probability 0.450.45. An undecided will switch to humanities or science with probabilities of 0.300.30 and 0.400.40, respectively. Complete parts (a) and (b) below. Question content area bottom Part 1 (a) Find the long-range prediction for the fraction of students in each of these three majors. First, find the transition matrix P. Let the first state be that a student is a humanities major, the second that a student is a science major, and the third state that a student is undecided. Pequals=Start 3 By 3 Table 1st Row 1st Column 0.35 2nd Column 0.40 3rd Column 0.25 2nd Row 1st Column 0.35 2nd Column 0.20 3rd Column 0.45 3rd Row 1st Column 0.30 2nd Column 0.40 3rd Column 0.30 EndTable 0.35 0.40 0.25 0.35 0.20 0.45 0.30 0.40 0.30 (Type an integer or decimal for each matrix element.) Part 2 Find the long-range prediction for the fraction of students in each of these three majors. The long-range prediction is enter your response here humanities, enter your response here science, and enter your response here undecided. (Type integers or simplified fractions.)
Solution
Part 1: Transition Matrix
The problem provides the probabilities for transitioning between the three states: humanities (H), science (S), and undecided (U). Using this, we construct the transition matrix :
\begin{bmatrix} 0.35 & 0.40 & 0.25 \\ 0.35 & 0.20 & 0.45 \\ 0.30 & 0.40 & 0.30 \end{bmatrix}$$ - Each row represents the current state (H, S, U), and each column represents the next state. - Rows sum to 1, ensuring that all probabilities are accounted for. ### Part 2: Long-Range Prediction The long-range prediction involves finding the **steady-state vector $$\pi$$:** $$\pi P = \pi$$ This implies solving the system of linear equations: 1. $$0.35\pi_1 + 0.35\pi_2 + 0.30\pi_3 = \pi_1$$ 2. $$0.40\pi_1 + 0.20\pi_2 + 0.40\pi_3 = \pi_2$$ 3. $$0.25\pi_1 + 0.45\pi_2 + 0.30\pi_3 = \pi_3$$ 4. $$\pi_1 + \pi_2 + \pi_3 = 1$$ (normalization) ### Solving the Equations We can rewrite the equations from the matrix form and solve them. Let me compute the steady-state vector $$\pi$$. numpy as np # Transition matrix P = np.array([ [0.35, 0.40, 0.25], [0.35, 0.20, 0.45], [0.30, 0.40, 0.30] ]) # Adding the normalization constraint: pi1 + pi2 + pi3 = 1 A = np.vstack([P.T - np.eye(3), np.ones(3)]) b = np.array([0, 0, 0, 1]) # Steady-state condition + normalization # Solving for steady-state probabilities steady_state = np.linalg.lstsq(A, b, rcond=None)[0] steady_state### Long-Range Prediction The steady-state probabilities are: - **Humanities**: $$\pi_1 = \frac{1}{3} \approx 0.333$$ - **Science**: $$\pi_2 = \frac{1}{3} \approx 0.333$$ - **Undecided**: $$\pi_3 = \frac{1}{3} \approx 0.333$$ This means, in the long run, the student population is evenly distributed among the three categories. --- Do you want a deeper explanation or further breakdown of the calculations? ### Related Questions 1. How does the transition matrix change if probabilities for certain transitions are adjusted? 2. What is the mathematical significance of the steady-state vector? 3. How can we interpret the symmetry in the transition matrix for this problem? 4. What happens if we remove one of the states (e.g., undecided)? 5. Can we use a computational approach to simulate long-term behavior for validation? ### Tip: When constructing a transition matrix, ensure that all rows sum to 1 to maintain proper probability distribution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability
Markov Chains
Linear Algebra
Formulas
Transition matrix representation: P = [P(i,j)], where P(i,j) is the probability of transitioning from state i to state j.
Steady-state vector: πP = π
Normalization condition: π1 + π2 + π3 = 1
Theorems
Steady-State Theorem for Markov Chains
Matrix Multiplication Properties
Suitable Grade Level
Grades 11-12 or Undergraduate
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