Question #41841

Let $A, B,$ and $C$ be three independent events associated with a random experiment, having probabilities $p, q,$ and $r$ respectively. Define two matrices $M_1$ and $M_2$ as follows: $M_1 = \begin{pmatrix} P(A) & P(B) & P(C) \\ P(A^c) & P(B^c) & P(C^c) \\ 1 & 1 & 1 \end{pmatrix}, \quad M_2 = \begin{pmatrix} P(B) & P(C) & P(A) \\ P(B^c) & P(C^c) & P(A^c) \\ 1 & 1 & 1 \end{pmatrix}$ It is given that the sum of the non-zero eigenvalues of $M_1$ is $\frac{13}{6}$, and the product of the non-zero eigenvalues of $M_2$ is $\frac{1}{6}$. If the expected number of events that occur during a single trial of the experiment is $\frac{13}{12}$, what is the probability that at least one of the events occurs?