# Probability On Numbers

Each of 20 identical cards is numbered with exactly one of the numbers 1,2,3,.....20. One card is drawn randomly and it is known that the number on the card is less than 13. What is the probability that the number on the card is an even number?

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That's just as if you had started with numbers $1, 2, \ldots, 12$ righ taway. – Hagen von Eitzen Sep 18 '12 at 17:28

If the card's value is less than 13, then given the other information its value must be in the range 1..12. This range has an even number of elements (12), and thus exactly half of them (6) will be even, so the probability of the card having an even value is $\dfrac{6}{12} = \dfrac{1}{2} =.5$.
For this problem, the approach above is clear and quick. One can also complicate things. Let $L$ be the event "less than $13$" and $E$ the event "even". We want $\Pr(E|L)$ (probability of $E$ given $L$). Then one can use the basic formula $\Pr(X|Y)=\frac{\Pr(X\cap Y)}{\Pr(Y)}$. For more complicated problems, this "conditional probability" approach is very useful, but it would be gross overkill here. – André Nicolas Sep 18 '12 at 18:04