Question

A pair of fair dice having six faces numbered from 1 to 6 are thrown once, suppose two events E and F are defined as -
E : Product of the two numbers appearing is divisible by 5.
F : At least one of the dice shows up the face one.
Then the events E and F are

• A. mutually exclusive
• B. independent
• C. neither independent nor mutually exclusive
• D. are equiprobable

Answer 1

Answer: (C), (D)

The correct options are
C neither independent nor mutually exclusive
D are equiprobable

Given, a pair of dices thrown once
$\Rightarrow$ total no. of outcomes $=36$
E: Product of two number appearing on dice is to be divisible by $5$.
No. of Outcomes$=(1,5),(5,1),(2,5),(5,2),(3,5),(5,3),(4,5),(5,4),(5,5),(5,6),(6,5)$
$P\left(E\right)=\frac{11}{36}$
F: Atleast one of dice shows the face 1.
No. of Outcomes$=(1,1),(1,2),(2,1),(1,3),(3,1),(1,4),(4,1),(1,5),(5,1),(1,6),(6,1)$
$P\left(F\right)=\frac{11}{36}$
and $P(E\cap F)=\frac{2}{36}=\frac{1}{18}$
since $P(E\cap F)\ne 0$
$\Rightarrow$ $E,F$ are not mutually exclusive and $P(E\cap F)$ is not equal to $P\left(E\right).P\left(F\right)$.
$\Rightarrow$ $E,F$ are not independent.
Since $P\left(E\right)=P\left(F\right)$$\Rightarrow$ $E,F$ are equiprobable too.