Question

Consider two pair of equations:
2x+y-5=0
ax+by-12=0
Find the probability such that given pair of equations represent parallel lines, where value of a and b is decided by rolling a pair of dice simultaneously.

• A. $\frac{1}{6}$
• B. $\frac{5}{6}$
• C. $\frac{7}{36}$
• D. $\frac{1}{12}$

Answer 1

The correct option is D $\frac{1}{12}$

Let 'E' be the event of system of equations being parallel in nature.
Now, consider the system of equations:
$a_{1}x+b_{1}y+c_1=0$
$a_{2}x+b_{2}y+c_2=0$
The above syetem of equations are parallel when $\frac{a_1}{a_2}$=$\frac{b_1}{b_2}$.

Substituting the value of $a_1$, $a_2$, $b_1$ and $b_2$ in above equation:
$\Rightarrow$ $\frac{2}{a}$=$\frac{1}{b}$
$\Rightarrow$ $\frac{a}{b}$=$\frac{2}{1}$

Favourable outcomes$(\text{where}\frac{a}{b}=2)=\left\{\right(2,1),(4,2),(6,3\left)\right\}$

$\therefore$Number of favorable outcomes for system of equation being parallel in nature = 3

For an event $E,$
$\text{Probability P(E)}$ = $\frac{\text{Number of outcomes favourable to E}}{\text{Number of all possible outcomes of the experiment}}$

P(system of equation are parallel)
$=\frac{3}{36}$ $=\frac{1}{12}$