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Suppose you have two cards: one is painted yellow on both sides and the other is painted yellow on one side and red on the other You select a card at random and view one side.

You notice it is yellow. What is the probability the other side is yellow?

dolfi224
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Let $YY$ denote the card which is yellow on both sides and $YR$ the one which is yellow on one side and red on the other.

We calculate the conditional probability $\mathbb{P}(YY|\textrm{observed one side yellow}) = \frac{\mathbb{P}(YY \textrm{ and observed one side yellow})}{\mathbb{P}(\textrm{observed one side yellow})}$

Note that $\mathbb{P}(YY \textrm{ and observed one side yellow}) = \frac{1}{2}$ since we select a card at random and $\mathbb{P}(\textrm{observed one side yellow}) = \frac{3}{4}$ since there are four sides of which three are yellow, so putting these facts together we get $$\mathbb{P}(YY|\textrm{observed one side yellow}) = \frac{2}{3}$$

ODF
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