Question

12 cards, numbered 1 to 12, are placed in a box, mixed up thoroughly and then a card is drawn at random from the box. If it is known that the number on the drawn card is more than 3, find the probability that it is an even number.

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Solution

Let A be the event ‘the number on the card drawn is even’ and B be the event ‘the number on the card is greater than 3’.

therefore space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight S space equals space open curly brackets 1 comma space 2 comma space 3 comma space 4 comma space 5 comma space 6 comma space 7 comma space 8 comma space 9 comma space 10 comma space 11 comma space 12 close curly brackets therefore space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight A space equals space open curly brackets 2 comma space 4 comma space 6 comma space 8 comma space 10 comma space 12 close curly brackets comma space straight B space equals space open curly brackets 4 comma space 5 comma space 6 comma space 7 comma space 8 comma space 9 comma space 10 comma space 11 comma space 12 close curly brackets

and

straight A space intersection thin space straight B space equals space open curly brackets 4 comma space 6 comma space 8 comma space 10 comma space 12 close curly brackets

therefore space space space space space space space straight P left parenthesis straight A right parenthesis space equals space 6 over 12 comma space space straight P left parenthesis straight B right parenthesis space equals space 9 over 12 space space and space space straight P left parenthesis straight A intersection straight B right parenthesis space equals space 5 over 12

Required probability =

straight P left parenthesis straight A space left enclose straight B right parenthesis end enclose space equals space fraction numerator straight P left parenthesis straight A intersection straight B right parenthesis over denominator straight P left parenthesis straight B right parenthesis end fraction space equals fraction numerator begin display style 5 over 12 end style over denominator begin display style 9 over 12 end style end fraction space equals 5 over 9

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