Question:
In a laboratory experiment, cats choose between foods A and B. If a cat had A, it will prefer A next time with probability $0.7$; if it had B, it will prefer A next time with probability $0.5$. The experiment is repeated under identical conditions. If $40%$ cats had A in the first experiment, then the percentage (rounded off to one decimal place) of cats that will prefer A in the third experiment is __________.
In a laboratory experiment, cats choose between foods A and B. If a cat had A, it will prefer A next time with probability $0.7$; if it had B, it will prefer A next time with probability $0.5$. The experiment is repeated under identical conditions. If $40%$ cats had A in the first experiment, then the percentage (rounded off to one decimal place) of cats that will prefer A in the third experiment is __________.
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With repeated behavior depending only on the previous choice, model the proportions by a two–state Markov chain and iterate the linear recursion.
Updated On: Dec 15, 2025
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Correct Answer: 61.5
Solution and Explanation
Let $p_n$ be the proportion that choose A in experiment $n$.
Transition (two–state Markov chain):
\[
p_{n+1}=0.7\,p_n+0.5(1-p_n)=0.5+0.2\,p_n .
\]
Given $p_1=0.4$,
\[
p_2=0.5+0.2(0.4)=0.58,\qquad
p_3=0.5+0.2(0.58)=0.616 .
\]
Thus the required percentage is $100\times 0.616= \boxed{61.6%}$.
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