Respuesta :
Given:
Two dice are thrown.
[tex]E_1[/tex] is the event that the sum of their dots is a prime number
[tex]E_2[/tex] is the event that 5 is the dot on the top of second die.
To find:
Whether [tex]P(E_1\cap E_2)=P(E_1)\cdot P(E_2)[/tex] 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).
[tex]E_1[/tex] is the event that the sum of their dots is a prime number.
[tex]E_1=\{(1,1),(1,2),(1,4),(1,6),(2,1),(2,3),(2,5),(3,2),(3,4),(4,1),(4,3),(5,2),(5,6),(6,1),(6,5)\}[/tex]
[tex]P(E_1)=\dfrac{15}{36}[/tex]
[tex]P(E_1)=\dfrac{5}{12}[/tex]
[tex]E_2[/tex] is the event that 5 is the dot on the top of second die.
[tex]E_2=\{(1,5), (2,5),(3,5),(4,5),(5,5),(6,5)\}[/tex]
[tex]P(E_2)=\dfrac{6}{36}[/tex]
[tex]P(E_2)=\dfrac{1}{6}[/tex]
The intersection of these two events is:
[tex]E_1\cap E_2=\{(2,5),(6,5)\}[/tex]
[tex]P(E_1\cap E_2)=\dfrac{2}{36}[/tex]
[tex]P(E_1\cap E_2)=\dfrac{1}{18}[/tex]
Now,
[tex]P(E_1)\cdot P(E_2)=\dfrac{5}{12}\cdot \dfrac{1}{6}[/tex]
[tex]P(E_1)\cdot P(E_2)=\dfrac{5}{72}[/tex]
[tex]P(E_1)\cdot P(E_2)\neq P(E_1\cap E_2)[/tex]
Therefore, the given statement is false because [tex]P(E_1\cap E_2)\neq P(E_1)\cdot P(E_2)[/tex].