Two fair dice are tossed, and the following events are defined: A: {Sum of the numbers showing is odd}
B: {Sum of the numbers showing is 9,11, or 12}
Are events A and B independent? Why?

A and B are not independent.

Step-by-step explanation:

Two events are said to be independent if :

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)

We can see from the above that:

Therefore, events A and B are not independent.

solution:

→c= a b , that is product of a and b is possible only when when a is m × n matrix and b is n × p matrix.

that is, number of columns of a= number of rows of b.

so, size of matrix , c= a b is , if a = (m × n)matrix

b= (n × p)matrix

then a b = [ (m × n)matrix] ×[   (n × p)matrix ]

→→c= a b= [m × p ] matrix

→→size of product a b will be equal to = number of rows of a × number of columns of b