An unbiased die is thrown twice. Let the event A be ‘odd number on the first throw’ and B the event ‘odd number on the second throw’. Check the independence of the events A and B.

straight S space equals space open curly brackets table row cell left parenthesis 1 comma space 1 right parenthesis end cell cell left parenthesis 1 comma space 2 right parenthesis end cell cell left parenthesis 1 comma space 3 right parenthesis end cell cell left parenthesis 1 comma space 4 right parenthesis end cell cell left parenthesis 1 comma space 5 right parenthesis end cell cell left parenthesis 1 comma space 6 right parenthesis end cell row cell left parenthesis 2 comma 1 right parenthesis end cell cell left parenthesis 2 comma space 2 right parenthesis end cell cell left parenthesis 2 comma space 3 right parenthesis end cell cell left parenthesis 2 comma space 4 right parenthesis end cell cell left parenthesis 2 comma space 5 right parenthesis end cell cell left parenthesis 2 comma space 6 right parenthesis end cell row cell left parenthesis 3 comma space 1 right parenthesis end cell cell left parenthesis 3 comma space 2 right parenthesis end cell cell left parenthesis 3 comma space 3 right parenthesis end cell cell left parenthesis 3 comma space 4 right parenthesis end cell cell left parenthesis 3 comma space 5 right parenthesis end cell cell left parenthesis 3 comma space 6 right parenthesis end cell row cell left parenthesis 4 comma space 1 right parenthesis end cell cell left parenthesis 4 comma space 2 right parenthesis end cell cell left parenthesis 4 comma space 3 right parenthesis end cell cell left parenthesis 4 comma space 4 right parenthesis end cell cell left parenthesis 4 comma space 5 right parenthesis end cell cell left parenthesis 4 comma space 6 right parenthesis end cell row cell left parenthesis 5 comma space 1 right parenthesis end cell cell left parenthesis 5 comma space 2 right parenthesis end cell cell left parenthesis 5 comma space 3 right parenthesis end cell cell left parenthesis 5 comma space 4 right parenthesis end cell cell left parenthesis 5 comma space 5 right parenthesis end cell cell left parenthesis 5 comma space 6 right parenthesis end cell row cell left parenthesis 6 comma space 1 right parenthesis end cell cell left parenthesis 6 comma space 2 right parenthesis end cell cell left parenthesis 6 comma space 3 right parenthesis end cell cell left parenthesis 6 comma space 4 right parenthesis end cell cell left parenthesis 6 comma space 5 right parenthesis end cell cell left parenthesis 6 comma space 6 right parenthesis end cell end table close curly brackets

Number of elements (outcomes) of the above example space is 6 x 6 = 36.

straight P left parenthesis straight A right parenthesis space equals 18 over 36 space equals 1 half space space and space straight P left parenthesis straight B right parenthesis space equals space 18 over 36 space equals 1 half

Also comma space straight P left parenthesis straight A intersection straight B right parenthesis space equals space straight P thin space left parenthesis odd space number space on space both space throws right parenthesis space space space space space space space space space space space space space space space space space equals space 9 over 36 space equals space 1 fourth

Now,

straight P left parenthesis straight A right parenthesis. space straight P left parenthesis straight B right parenthesis space equals space 1 half cross times 1 half space equals space 1 fourth

Clearly,

straight P left parenthesis straight A space intersection space straight B right parenthesis space equals space straight P left parenthesis straight A right parenthesis space cross times space straight P left parenthesis straight B right parenthesis

∴ A and B are independent events.

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