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Question
The subscription department of a magazine sends out a letter to a large mailing list inviting subscriptions for the magazine. Some of the people receiving this letter already subscribe to the magazine while others do not. From this mailing list, 60% of those who already subscribe will subscribe again while 25% of those who do not now subscribe will subscribe. On the last letter it was found that 40% of those receiving it ordered a subscription. What percent of those receiving the current letter can be expected to order a subscription?
Solution
Let X represent people who subscribe for the magazine and Y represent people who do not subscribe for the magazine. Then
(X X) ⇒ those who already subscribed will do it again.
(X Y) ⇒ those who already subscribed will not do it again.
(Y X) ⇒ those who have not subscribed will do it now.
(Y Y) ⇒ those who have not subscribed already will not do it now also.
From the question,
(X X) = 60% = `60/100` = 0.6
(Y Y) = 1 – 0.6 = 0.4
(Y X) = 25% = `25/100` = 0.25
(Y Y) = 1 – 0.25 = 0.75
The current position is given By X = 40% and Y = 60%
`(("X", "Y"),(0.4, 0.6))`
The transition probability matrix is given by \[{\begin{matrix} & \begin{matrix}\text{X}&&\text{Y}\end{matrix} \\ T=\begin{matrix}\text{X}\\\text{Y}\end{matrix} & \begin{pmatrix}0.6&0.4\\0.25&0.75\end{pmatrix}\\ \end{matrix}}\]
We have to predict the value of X and Y after the current letter is sent.
\[\begin{matrix} & \begin{matrix}\text{X}&&\text{Y}\end{matrix} \\ & \begin{pmatrix}0.4&0.6\end{pmatrix}\\&&& \end{matrix} \begin{matrix}\phantom{..} \begin{matrix}\text{X}&&\text{Y}\end{matrix} \\ \begin{matrix}\text{X}\\\text{Y}\end{matrix} \begin{pmatrix}0.6&0.4\\0.25&0.75\end{pmatrix}\\ \end{matrix}\]
`(("X","Y"),(0.24 + 0.15, 0.16 + 0.45)) = ((0.39, 0.61))`
i.e., X = 0.39 = `0.39/100 xx 100` = 39%
Y = 0.61 = `0.61/100 xx 100` = 61%
Thus 39% of those receiving the current letter can be expected to order a subscription.
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