A bag has $3$ red and $5$ blue balls. Two balls are drawn one after the other WITHOUT replacement. What is…

A bag has $3$ red and $5$ blue balls. Two balls are drawn one after the other WITHOUT replacement. What is the probability that both are blue?
  1. $\dfrac{5}{14}$
  2. $\dfrac{25}{64}$
  3. $\dfrac{3}{8}$
  4. $\dfrac{4}{7}$

Solution

$P(\text{1st blue}) = \dfrac{5}{8}$, $P(\text{2nd blue} \mid \text{1st blue}) = \dfrac{4}{7}$. $P(\text{both blue}) = \dfrac{5}{8} \times \dfrac{4}{7} = \dfrac{20}{56} = \dfrac{5}{14}$.

Asked in: IMO

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