WXXX, WYYY, AND WZZZ are three radio stations that compete for listeners. Each m

Question

WXXX, WYYY, AND WZZZ are three radio stations that compete for listeners. Each month some of the listeners switch from one station to another while some remain loyal to their current station.

It has been found that each month:

20% of the listeners of WXXX switch to WYYY
10% of the listeners of WXXX switch to WZZZ
30% of the listeners of WYYY switch to WXXX
10% of the listeners of WYYY switch to WZZZ
20% of the listeners of WZZZ switch to WXXX
30% of the listeners of WZZZ switch to WYYY

Currently, the listening audience is divided as follows:

15% listen to WXXX
40% listen to WYYY
45% listen to WZZZ

Use this information to answer the following questions. (Give your answers correct to three decimal places. Note that you are asked to give decimal values and not percentages. For example, you would need to enter 0.628 as opposed to 62.800%.)

(a) What fraction of the listeners will WZZZ have after 2 months?

(b) What fraction of the listeners will WXXX have after 3 months?

(c) What fraction of the listeners will WYYY have after 5 months?

Explanation / Answer

Let us assume there are 1000 listeners in total.

Currently, there are 150 listeners of WXXX, 400 listeners of WYYY. and 450 listeners of WZZZ.

a.) After one month, WZZZ will gain 10% of WXXX listeners and WYYY listeners and lose 50% listeners in total.

Lost Listeners = 225

Gained listeners = 15 + 40 = 55

WZZZ listeners after one month = 450 - 225 + 55 = 280

Now, WYYY loses 40% of its listeners and gains 20% of WXXX listeners and 30% of WZZZ listeners

WYYY listeners after one month = 400*0.6 + 0.2*150 + 0.3*450 = 405

Now, WXXX loses 30% of its listeners and gains 30% of WYYY listeners and 20% of WZZZ listeners

WXXX listeners after one month = 150*0.7 + 0.3*400 + 0.2*450 = 315

Now after two month for WZZZ = 280*0.5 + 0.1*315 + 0.1*405 = 212

Fraction of the listeners WZZZ will have after 2 months = 0.212

b.) & c.) Similar to what is done in previous part

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