Probability
Three independent events; determinant of probability matrix
MMTS_Full_Test_07
Grade 12
Question:
Let $E_1, E_2, E_3$ be three independent events such that $3P(E_1\cap\bar{E_2}\cap\bar{E_3})=P(\bar{E_1}\cap E_2\cap\bar{E_3})=9P(\bar{E_1}\cap\bar{E_2}\cap E_3)=3-3P(E_1\cup E_2\cup E_3)$. If the absolute value of $\begin{vmatrix}P(E_1)&P(E_2)&P(E_3)\\P(E_2)&P(E_3)&P(E_1)\\P(E_3)&P(E_1)&P(E_2)\end{vmatrix}=\dfrac{a}{b}$ where $a,b\in\mathbb{N}$, then least value of $a+b$ is
Step-by-Step Solution
Key Concept: Let $P(E_i)=x,y,z$. Translate the conditions into equations using independence. Solve the system to find $x,y,z$, then compute the determinant.
$x=1/2$, $y=3/4$, $z=1/4$. $|\text{det}|=9/32$. $a+b=9+32=41$.
Correct Answer: 41