Question

Question 3: On any given day, a student is either healthy or ill. Of the students who are healthy today, 95% will be healthy tomorrow. Of the students who are ill today, 55% will still be ill tomorrow.

a. What is the stochastic matrix for this situation?

b. Suppose 20% of the students are ill on Monday. What fraction or percentage of the students are likely to be ill on Tuesday? On Wednesday?

c. If a student is well today, what is the probability that he or she will be well two days from now?

Short answer

1. The stochastic matrix is \(P = \left( {\begin{array}{*{20}{c}}{.95}&{.45}\\{.05}&{.55}\end{array}} \right)\).
2. 15% of the students are ill on Tuesday, and 12.5% of the students are ill on Wednesday.

3. The probability that the students will be well two days from now is 0.925.

Step-by-step solution

Determine the stochastic matrix

a)

Let \(H\) represent a healthy student and \(I\) represent an ill student.

The following table represents the student’s conditions.

From:		To:
H	I
.95	.45	H
.05	.55	I

Therefore, the stochastic matrix is \(P = \left( {\begin{array}{*{20}{c}}{.95}&{.45}\\{.05}&{.55}\end{array}} \right)\).

Determine the percentage of students who are likely to be ill on Tuesday

b)

Theorem 18states that if \(P\) is a \(n \times n\) regular stochastic matrix,then \(P\) has a unique steady-state vector \({\mathop{\rm q}\nolimits} \).Further, if \({{\mathop{\rm x}\nolimits} _0}\) is any initial state and \({{\mathop{\rm x}\nolimits} _{k + 1}} = P{{\mathop{\rm x}\nolimits} _k}\) for \(k = 0,1,2,....\) then the Markov chain \(\left\{ {{{\mathop{\rm x}\nolimits} _k}} \right\}\) converges to \({\mathop{\rm q}\nolimits} \)as \(k \to \infty \).

The initial state vector is \({{\mathop{\rm x}\nolimits} _0} = \left( {\begin{array}{*{20}{c}}{.8}\\{.2}\end{array}} \right)\) because 20% of the students are ill on Monday.

Compute \({{\mathop{\rm x}\nolimits} _1}\) to find Tuesday’s percentage.

\(\begin{array}{c}{{\mathop{\rm x}\nolimits} _1} = P{{\mathop{\rm x}\nolimits} _0}\\ = \left( {\begin{array}{*{20}{c}}{.95}&{.45}\\{.05}&{.55}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{.8}\\{.2}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.76 + 0.09}\\{0.04 + 0.11}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.85}\\{0.15}\end{array}} \right)\end{array}\)

Thus, 15% of the students are ill on Tuesday.

Determine the percentage of students likely to be ill on Wednesday

Compute \({{\mathop{\rm x}\nolimits} _1}\) to find Tuesday’s percentage.

\(\begin{array}{c}{{\mathop{\rm x}\nolimits} _2} = P{{\mathop{\rm x}\nolimits} _1}\\ = \left( {\begin{array}{*{20}{c}}{.95}&{.45}\\{.05}&{.55}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{0.85}\\{0.15}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.8075 + 0.0675}\\{0.0425 + 0.0825}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.875}\\{0.125}\end{array}} \right)\end{array}\)

Hence, 12.5% of the students are ill on Wednesday.

Determine the probability that the student will be well in two days

c)

Suppose the student is well today, then the initial state vector is \({{\mathop{\rm x}\nolimits} _0} = \left( {\begin{array}{*{20}{c}}1\\0\end{array}} \right)\).

Compute \({{\mathop{\rm x}\nolimits} _2}\) as shown below:

\(\begin{array}{c}{{\mathop{\rm x}\nolimits} _1} = P{{\mathop{\rm x}\nolimits} _0}\\ = \left( {\begin{array}{*{20}{c}}{.95}&{.45}\\{.05}&{.55}\end{array}} \right)\left( {\begin{array}{*{20}{c}}1\\0\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.95 + 0}\\{0.05 + 0}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.95}\\{0.05}\end{array}} \right)\end{array}\)

\(\begin{array}{c}{{\mathop{\rm x}\nolimits} _2} = P{{\mathop{\rm x}\nolimits} _1}\\ = \left( {\begin{array}{*{20}{c}}{.95}&{.45}\\{.05}&{.55}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{0.95}\\{0.05}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.9025 + 0.0225}\\{0.0475 + 0.0275}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{0.925}\\{0.075}\end{array}} \right)\end{array}\)

Thus, the probability that the students will be well two days from now is 0.925.