+182 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 draws is labeled with an even number, and the second card Grok draws is labeled with a multiple of 3.
+403 There are two events we need to consider:
Event 1: Drawing an even number first.
There are 4 colors in the deck.
Each color has one even number (2, 4, or 6).
So, there are 4 * 3 = 12 cards that satisfy this condition (even number, any color).
Event 2: Drawing a multiple of 3 after drawing an even number (without replacement).
After drawing the first card, there are only 27 cards remaining.
There are 3 multiples of 3 left (3 and 6, since one even number - 6 - is already drawn).
However, since Grok isn't replacing the first card, there are only 2 colors left that have multiples of 3 (red and blue, as the even number 6 was likely green or yellow).
Therefore, there are 2 * 1 = 2 cards that satisfy this condition (multiple of 3, remaining colors).
Total Favorable Cases:
To get the probability, we need the number of favorable cases (both events happening) divided by the total number of possible cases (drawing any two cards).
Favorable cases: 12 (Event 1) * 2 (Event 2) = 24
Total possible cases: There are 28 cards total (7 numbers * 4 colors), and we draw 2 without replacement. So, the total number of possible choices is 28C2 (28 choose 2) which is 28 * 27 / (2 * 1) = 378
Probability:
Therefore, the probability that Professor Grok draws an even number first, followed by a multiple of 3 (without replacement), is:
Probability = Favorable Cases / Total Possible Cases Probability = 24 / 378
Simplifying the fraction:
Both the numerator (24) and denominator (378) have a common divisor of 6. We can simplify:
Probability = (24 / 6) / (378 / 6) Probability = 4 / 63
So, the probability is 4/63.
