Question

A college dormitory houses 200 students. Those who watch an hour or more of television on any day always watch for less than an hour the next day. One-fourth of those who watch television for less than an hour one day will watch an hour or more the next day. Half of the students watched television for an hour or more today. How many will watch television for an hour or more tomorrow, in 2 days, and in 40 days

Answer 1

Answer:

Tomorrow = 25 students

In 2 days = 44 students

In 40 days = 40 students

Step-by-step explanation:

Let A =  watch more than an hour

     B = watch less than an hour

First of all we need to make a transition matrix

$P = \begin{bmatrix}0 & 1\\ 0.25 & 0.75\end{bmatrix} \binom{A}{B}$

Also we are given,

X = [100, 100]

       A       B

Therefore,

$XP=\begin{bmatrix}100 & 100\end{bmatrix} \times \begin{bmatrix}0 & 1\\ 0.25 & 0.75\end{bmatrix}$

     = [25, 175]

Thus, 25 students will watch television for an hour or more tomorrow.

Now for 2 days,

$XP^2=\begin{bmatrix}100 & 100\end{bmatrix} \times \begin{bmatrix}0 & 1\\ 0.25 & 0.75\end{bmatrix} \times \begin{bmatrix}0 & 1\\ 0.25 & 0.75\end{bmatrix}$

       = [43.75, 156.25]

Hence, 44 students will watch television for an hour or more in 2 days.

Now in 40 days,

$XP^40=\begin{bmatrix}100 & 100\end{bmatrix} \times \begin{bmatrix}0 & 1\\ 0.25 & 0.75\end{bmatrix}^{40}$

But instead of multiplying the above matrix 40 times, we can we can find the long term division method.

Let

x =  long term distribution of students watching television >1 hour

y =   long term distribution of students watching television <1 hour

Therefore,

$\begin{bmatrix}x & y\end{bmatrix} \times \begin{bmatrix}0 & 1\\ 0.25 & 0.75\end{bmatrix} = \begin{bmatrix}x & y\end{bmatrix}$

⇒ 0.25 y = x

And we know that x + y = 200

∴ 0.25 y + y = 200

∴ y = 160

And x = 40 ( ∵ x + y = 200)

Thus 40 students will watch televisions in the 40 days for an hour or more.