A company entered into a new market (sells product X) and started operating. In the first month of operation, it is found that 25% of customers are using the product X and while 75% of people still using other substitutes of X.
The company has it's own records as follows: (from other cities or etc.)
If somebody uses product X in current month, for him, there is 80% chance of keep using X in the next month, and 20% chance of switching to other substitute.
If somebody not using product X in current month, for him, there is 30% chance of using X in the next month, and 70% chance of not changing to X.
Find the % of customers using the product X after a long time.
My Aproach/Attempt:
Let us define abbreviations for the following:
Using the product X $\Rightarrow$ UPX
Using the the other Substitutes of X $\Rightarrow$ OSX
Keep using the product X in the next month $\Rightarrow$ KUPMX
Switch to other product $\Rightarrow$ SWOP
Given
$P\left ( UPX \right )=0.25$,$P\left ( KUPMX/UPX \right )=0.80$,
$P\left (SWOP/ UPX \right )=0.20$,$P\left ( KPX/\left ( UPX \right ){}' \right )=0.30$
Given question can be simplified to
$P\left ( UPX/KUPMX \right )=\left ( P\left ( KUPMX/UPX \right )*P\left ( UPX \right ) \right )/\left ( P\left ( KUPMX/UPX \right )*P\left ( UPX \right ) \right )+\left ( P\left ( KUPMX/\left ( UPX \right ){}' \right )*P\left ( UPX \right ) {}'\right )$
=$0.80*0.25/\left ( 0.80*0.25 \right )+\left ( 0.30*0.75 \right )=.4705$
=$47$%
Am I correct?
Let $U_n$ be the event that a person uses the product in the $n$th month and let $\overline{U_n}$ be the complement. Then
\begin{align} P(U_{n+1})&=P(U_{n+1}|U_n)P(U_n)+P(U_{n+1}|\overline{U_n})P(\overline{U_n})\\ &=0.8P(U_n)+0.3P(\overline{U_n})\\ &=0.8P(U_n)+0.3(1-P(U_n)), \end{align}
with $P(U_1)=0.25,$ which is a linear recursion. It can be solved to give $$P(U_n)=\frac{6-7\times2^{-n}}{10},$$ which, in the limit $n\rightarrow \infty$, gives $\frac{6}{10}=0.6$.