Suppose that we have 3 cards that are identical in form, except that both sides of the first card are colored red, both sides of the second card are colored black, and one side of the third card is colored red and the other side black. The 3 cards are mixed up in a hat, and 1 card is randomly selected and put down on the ground. If the upper side of the chosen card is colored red, the probability that the other side, colored black is m , then the value of 6m is _____

Question

Suppose that we have 3 cards that are identical in form, except that both sides of the first card are colored red, both sides of the second card are colored black, and one side of the third card is colored red and the other side black. The 3 cards are mixed up in a hat, and 1 card is randomly selected and put down on the ground. If the upper side of the chosen card is colored red, the probability that the other side, colored black is m  , then the value of  6m  is _____

Accepted Answer

Let RR, BB, and RB denote, respectively, the events that the chosen card is all red, all black, or the red-black card. Also, let R be the event that the upturned side of the chosen card is red. Then the desired probability is obtained by
$P (R B ∣ R) = \frac{P (R B \cap R)}{P (R)} = \frac{P (R ∣ R B) P (R B)}{P (R ∣ R R) P (R R) + P (R ∣ R B) P (R B) + P (R ∣ B B) P (B B)} = \frac{(\frac{1}{2}) (\frac{1}{3})}{(1) (\frac{1}{3}) + (\frac{1}{2}) (\frac{1}{3}) + 0 (\frac{1}{3})} = \frac{1}{3}$

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