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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 (1-7). Among all the cards of each color, there is exactly one card labeled with each number. The cards in Ms. Q's deck are shown below.

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 3

May 10, 2024

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To solve this problem, let's break it down step by step:

1. **Finding the probability of drawing an even number on the first draw:**

There are 7 numbers on each color, and 3 of them are even (2, 4, 6). So, the probability of drawing an even number on the first draw is $$\frac{3}{7}$$.

2. **Finding the probability of drawing a multiple of 3 on the second draw:**

There are still 7 numbers on each color, and 2 of them are multiples of 3 (3, 6). After the first draw, there are 13 cards remaining. Among these, there are 4 cards labeled with multiples of 3. So, the probability of drawing a multiple of 3 on the second draw is $$\frac{4}{13}$$.

3. **Multiplying the probabilities:**

The probability of drawing an even number on the first draw and a multiple of 3 on the second draw is the product of the individual probabilities:
$$\frac{3}{7} \times \frac{4}{13}$$

4. **Calculating the final probability:**

Multiply the fractions:

$$\frac{3}{7} \times \frac{4}{13} = \frac{12}{91}$$

So, 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 $$\frac{12}{91}$$.

May 13, 2024