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. 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 is red or green, and the second card Grok draws is odd and yellow?
There are a total of $4 \times 7 = 28$ cards in Ms. Q's deck.
First, we count the number of favorable outcomes.
There are $2$ red cards and $2$ green cards.
For the second card, there are $4$ odd cards and $7$ yellow cards. We need to subtract the $2$ odd yellow cards that were already counted in the cases when Grok drew a red card.
So, there are $2 \cdot (4 + 7 - 2) = 18$ favorable outcomes.
Next, we count the number of possible outcomes.
There are $2$ choices for the first card, and $27$ choices for the second card. Once the first card is drawn, there are only $27$ cards left in the deck.
Therefore, the probability is $\dfrac{2 \cdot 18}{2 \cdot 27} = \boxed{\dfrac{2}{3}}$.