AMC12 2021 A

AMC12 2021 A · Q6

AMC12 2021 A · Q6. It mainly tests Ratios & proportions, Probability (basic).

A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is $\frac13$. When $4$ black cards are added to the deck, the probability of choosing red becomes $\frac14$. How many cards were in the deck originally?

一副牌只有红牌和黑牌。随机抽取一张牌是红牌的概率为 $\frac13$。当向牌堆中加入 $4$ 张黑牌后，选择红牌的概率变为 $\frac14$。原牌堆中有多少张牌？

(A) 6 6

(B) 9 9

(D) 15 15

(E) 18 18

Answer

Correct choice: (C)

正确答案：（C）

Solution

If the probability of choosing a red card is $\frac{1}{3}$, the red and black cards are in ratio $1:2$. This means at the beginning there are $x$ red cards and $2x$ black cards. After $4$ black cards are added, there are $2x+4$ black cards. This time, the probability of choosing a red card is $\frac{1}{4}$ so the ratio of red to black cards is $1:3$. This means in the new deck the number of black cards is also $3x$ for the same $x$ red cards. So, $3x = 2x + 4$ and $x=4$ meaning there are $4$ red cards in the deck at the start and $2(4) = 8$ black cards. So, the answer is $8+4 = 12 = \boxed{\textbf{(C) }12}$.

如果选择红牌的概率为 $\frac{1}{3}$，则红牌和黑牌的比例为 $1:2$。这意味着最初有 $x$ 张红牌和 $2x$ 张黑牌。加入 $4$ 张黑牌后，黑牌总数为 $2x+4$ 张。此时，选择红牌的概率为 $\frac{1}{4}$，所以红牌与黑牌的比例为 $1:3$。这意味着在新牌堆中，黑牌数为 $3x$ 张（红牌仍为 $x$ 张）。因此，$3x = 2x + 4$，解得 $x=4$，最初有 $4$ 张红牌和 $2(4) = 8$ 张黑牌。所以，总数为 $8+4 = 12 = \boxed{\textbf{(C) }12}$。

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