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# probability

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In Ms. Q's deck of cards, every card is one of four colors (red, green, blue, and yellow), and is labeled with one of seven numbers ( 1, 2, 3, 4, 5, 6, and 7). Among all the cards of each color, there is exactly one card labeled with each number. Professor Grok draws two cards from Ms. Q's deck at random without replacement. What is the probability that the first card Grok draws is labeled with an even number, and the second card Grok draws is labeled with a multiple of

May 7, 2024

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To solve this problem, let's first find the total number of possible pairs of cards that Professor Grok can draw. Since there are 4 colors and 7 numbers, there are $$4 \times 7 = 28$$ cards in total.

Now, let's find the number of pairs where the first card drawn is labeled with an even number and the second card drawn is labeled with a multiple of 3.

1. There are 7 even numbers among 1 to 7: 2, 4, 6.

2. There are 7 multiples of 3 among 1 to 7: 3, 6.

To find the number of pairs with the desired properties:

- For the first card, there are 7 even numbers, and for the second card, there are 7 multiples of 3.

- So, the total number of pairs with the desired properties is $$7 \times 7 = 49$$.

Now, to find the probability, divide the number of pairs with the desired properties by the total number of possible pairs:

$\text{Probability} = \frac{\text{Number of pairs with desired properties}}{\text{Total number of possible pairs}} = \frac{49}{28 \times 27}$

However, since Professor Grok is drawing without replacement, after the first card is drawn, there are only 27 cards left. So, the denominator should be $$28 \times 27$$.

Thus, the probability that the first card Grok draws is labeled with an even number, and the second card Grok draws is labeled with a multiple of 3 is:

$\frac{49}{28 \times 27} = \frac{7}{108}.$

May 9, 2024