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A loaded gun remains loaded after its visit with probability $1/2$. An unloaded gun is loaded and then not fired with probability $(3/4)(1/2)=3/8$. The visited gun's new loaded probability is thereforePreservation requires $p^{\prime }=p$, givingThis is product stationarity under single-coordinate Markov updates: the visited coordinate has the same Bernoulli distribution afterward, and its independent local random decisions do not affect the other coordinates. Thus the originally independent indicator random variables remain jointly independent after each scheduled visit, not merely equal in their marginal probabilities.

Let $X_j$ be the indicator random variable that gun $j$ is loaded. Then $N={\sum }_{j=1}^m{X}_j$ has binomial distribution with parameters $m,3/7$. By linearity of expectation and independence of the indicator random variables,