Question

Two dice are thrown, 1 is the event that the sum of their

dots is a prime number and 2 is the event that 5 is the dot on
the top of second die. Check whether (1 ∩ 2) =
(1). (2)

Answer 1

Given:

Two dice are thrown.

`E_1` is the event that the sum of their dots is a prime number

`E_2` is the event that 5 is the dot on the top of second die.

To find:

Whether
`P(E_1\cap E_2)=P(E_1)\cdot P(E_2)` is true or false.

Solution:

If two dice thrown, then the total possible outcomes are:

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6),

(3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6),

(5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6).

`E_1` is the event that the sum of their dots is a prime number.

`P(E_1)=(15)/(36)`

`P(E_1)=(5)/(12)`

`E_2` is the event that 5 is the dot on the top of second die.

`E_2=\{(1,5), (2,5),(3,5),(4,5),(5,5),(6,5)\}`

`P(E_2)=(6)/(36)`

`P(E_2)=(1)/(6)`

The intersection of these two events is:

`E_1\cap E_2=\{(2,5),(6,5)\}`

`P(E_1\cap E_2)=(2)/(36)`

`P(E_1\cap E_2)=(1)/(18)`

Now,

`P(E_1)\cdot P(E_2)=(5)/(12)\cdot (1)/(6)`

`P(E_1)\cdot P(E_2)=(5)/(72)`

`P(E_1)\cdot P(E_2)\\eq P(E_1\cap E_2)`

Therefore, the given statement is false because
`P(E_1\cap E_2)\\eq P(E_1)\cdot P(E_2)`.