Example 16 As shown in Figure $12-4, J_{A} 、 J_{B} 、 J_{C} 、 J_{D}$ are the same type of automatic control switch. It is known that within a certain period, the probability of them closing normally is $p(0<p<1)$, and they are independent of each other. Find the probability that the indicator light in the circuit flashes during this period.
Given the problem, a basic event can be designed as a repeatable permutation of filling in “on” or “off” in the blanks ( , , , ). For example, (on, off, off, off) represents a situation where the four switches $J_{A}$, $J_{B}$, $J_{C}$, and $J_{D}$ are sequentially $J_{A}$ on, $J_{B}$ off, $J_{C}$ off, and $J_{D}$ off. Thus, the total number of basic events is $2^{4}=16$.

Let $J_{A}$, $J_{B}$, $J_{C}$, and $J_{D}$ being closed be denoted as events $A$, $B$, $C$, and $D$, respectively. Then, (on, off, off, off) corresponds to the event $\bar{A} B C D$. Since $\bar{A}$, $B$, $C$, and $D$ are mutually independent, the probability $P(\bar{A} B C D) = p^{3}(1-p)$, and $P(A B \bar{C} \bar{D}) = p^{2}(1-p)^{2}$. This indicates that when $p \neq \frac{1}{2}$, the probabilities of the 16 basic events are not all equal.
$$
\text{Let } \begin{aligned}
M & = \text{ “indicator light flashes”, then} \\
P(M) & = P(R+S) = P(A+B+S) \\
& = 1 - P(\overline{A B} \bar{S}) \\
& = 1 - P(\bar{A}) P(\bar{B}) P(\bar{S}) \\
& = 1 - (1-p)^{2}(1-p^{2}) \\
& = 2p - 2p^{3} + p^{4}.
\end{aligned}
$$