Of the three independent events E_1, E_2 and E_3, the probability… - Vidyadip

MCQ

Of the three independent events $E_1, E_2$ and $E_3$, the probability that only $E_1$ occurs is $\alpha$,only $E_2$ occurs is $\beta$ and only $E_3$ occurs is $\gamma$. Let the probability $p$ that none of events $E_1, E_2$ or $E_3$ occurs satisfy the equations ( $\alpha$ $-2 \beta) p=\alpha \beta$ and $(\beta-3 \gamma) p=2 \beta \gamma$. All the given probabilities are assumed to lie in the interval $(0,1)$.

Then $\frac{\text { Probability of occurrence of } E_1}{\text { Probability of occurrence of } E_3}=$

A
$5$
✓
$6$
C
$7$
D
$8$

✓

Answer

Correct option: B.

$6$

b
Let $x, y, z$ be probability of $E_1, E_2, E_3$ respectively

$\Rightarrow \quad x(1-y)(1-z)=\alpha $

$\Rightarrow \quad y(1-x)(1-z)=\beta $

$\Rightarrow \quad z(1-x)(1-y)=\gamma $

$\Rightarrow \quad (1-x)(1-y)(1-z)=P$

Putting in the given relation we get $x=2 y$ and $y=3 z$

$\Rightarrow \quad x=6 z \Rightarrow \frac{x}{z}=6$

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