A college dormitory houses 600 students. Those who watch an hour or more of tele

Question

A college dormitory houses 600 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 20 days? (Round to the nearest whole number.) (a) tomorrow 75 students (b) in 2 days students (c) in 20 days students

Explanation / Answer

(a) Half of the students, i.e. 300 students watch TV for an hour or more today. The probability/transition matrix is A =

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¼

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3/4

The number of students who will watch TV for an hour or more tomorrow is A(300,300)T = (75,525)T. Thus, 75 students will watch TV for an hour or more tomorrow.

(b) The number of students who will watch TV for an hour or more, in 2 days, is given by A2(300,300)T = A(75,525)T = (131.25,468.75)T or, (131,469)T ( on rounding off). Thus, 131 students will watch TV for an hour or more in 2 days.

(c ) The stady state vector is given by AX = X , where X = (x,y)T . Then, we have y/4 = x or, y = 4x. Also, we know that x+y = 600 so that x+4x = 600 or 5x = 600 so that x = 120 and y = 4*120 = 480. Thus, the number of students who will watch TV for an hour or more, in 20 days, is 120.

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3/4

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