Respuesta :
Answer:
A and B are not independent.
Step-by-step explanation:
Two events are said to be independent if :[tex]P(A \cap B)=P(A)P(B)[/tex]
A: {Sum of the numbers showing is odd}
B: {Sum of the numbers showing is 9,11, or 12}
Given two dice, the sample space is given as:
(1,1), (2,1), (3,1), (4,1), (5,1), (6,1)
(1,2), (2,2), (3,2), (4,2), (5,2), (6,2)
(1,3), (2,3), (3,3), (4,3), (5,3), (6,3)
(1,4), (2,4), (3,4), (4,4), (5,4), (6,4)
(1,5), (2,5), (3,5), (4,5), (5,5), (6,5)
(1,6), (2,6), (3,6), (4,6), (5,6), (6,6)
The outcomes of event A:{Sum of the numbers showing is odd} are:
(2,1), (4,1), (6,1) , (1,2), (3,2),(5,2), (2,3), (4,3), (6,3)
(1,4), (3,4), (5,4), (2,5), (4,5), (6,5) , (1,6), (3,6), (5,6)
P(A)=18/36
The outcomes of event B:{Sum of the numbers showing is 9,11, or 12} are:
(6,3) , (5,4),(4,5) (6,5) , (3,6), (5,6), (6,6)
P(B)=7/36
The intersection of A and B are:
(6,3) , (5,4),(4,5) (6,5) , (3,6), (5,6)
[tex]P(A\cap B)=6/36[/tex]
We can see from the above that:
[tex]P(A)P(B)=\dfrac{7}{36}\times \dfrac{18}{36}=\dfrac{7}{72}\neq P(A\cap B)[/tex]
Therefore, events A and B are not independent.
