A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event 'the number is even', and B be the event, 'the number is red'. Are A and B independent?

Solution

S = {1, 2, 3, 4, 5, 6}
Now A: 'number is even'
and B: 'number is red'
$therefore$ $straight A space equals space left curly bracket 2 comma space 4 comma space 6 right curly bracket space and space straight B space equals space left curly bracket 1 comma space 2 comma space 3 right curly bracket$
Also, $straight A intersection straight B space equals space left curly bracket 2 right curly bracket.$
Now, $straight P left parenthesis straight A right parenthesis space equals 3 over 6 space equals 1 half comma space space space straight P left parenthesis straight B right parenthesis space equals space 3 over 6 space equals space 1 half space space and space straight P left parenthesis straight A space intersection space straight B right parenthesis space equals space 1 over 6 space not equal to space 1 half cross times 1 half$
$therefore space space space space space straight P left parenthesis space straight A intersection straight B right parenthesis space not equal to space straight P left parenthesis straight A right parenthesis space straight P left parenthesis straight B right parenthesis$
∴ A and B are not independent.

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Probability

A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event 'the number is even', and B be the event, 'the number is red'. Are A and B independent?

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