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प्रश्न
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, 45% of those who already subscribe will subscribe again while 30% 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?
उत्तर
Let X represent people who subscribe for the magazine and Y represent persons who do not subscribe for the magazine.
Now there are four cases,
(X, X) ⇒ those who already subscribed will subscribe again.
(X, Y) ⇒ those who already subscribed will not subscribe again.
(Y, X) ⇒ those who have not subscribed will do it now.
(Y, Y) ⇒ those who have not subscribed will not do it now also.
From the problem, we can see that
(X, X) = 45% = 0.45
(X, Y) = 100 – 45 = 55% = 0.55
(Y, X) = 30% = 0.3
(Y, Y) = (100 – 30) = 70% = 0.7
The transition probability matrix is given by
\[{\begin{matrix} & \begin{matrix}X&&Y\end{matrix} \\ T=\begin{matrix}X\\Y\end{matrix} & \begin{pmatrix}0.45&0.55\\0.3&0.7\end{pmatrix}\\ \end{matrix}}\]
The values of X and Y are given as
X = 40% = 0.4
Y = (100 – 40) = 60% = 0.6
We have to predict the value of X and Y after the current letter is sent.
It is done as below
\[\begin{matrix} & \begin{matrix}X&&Y\end{matrix} \\ & \begin{pmatrix}0.4&0.6\end{pmatrix}\\&&& \end{matrix} \begin{matrix} & \begin{matrix}X&&Y\end{matrix} \\ \begin{matrix}X\\Y\end{matrix} & \begin{pmatrix}0.45&0.55\\0.3&0.7\end{pmatrix}\\ \end{matrix}\]
= (0.4 × 0.45 + 0.6 × 0.3, 0.4 × 0.55 + 0.6 × 0.7)
= (0.18 + 0.18, 0.22 + 0.42)
= (036, 0.64)
= `(("X", "Y"),(036, 0.64))`
That is X = 36% and Y = 64%
Thus 36% of those receiving the current letter can be expected to order a subscription.
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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?
