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 has an even number, and the second card Grok draws has a multiple of 3?
The probability of event A happening, then event B, is the probability of event A happening times the probability of event B happening given that event A already happened. In this case, event A is drawing an even number and event B is drawing a multiple of 3.
There are 3 even numbers in the deck (2, 4, and 6) and 3 multiples of 3 (3, 6, and 9). So, the probability of drawing an even number is 3/7.
Once Grok has drawn an even number, there are only 2 multiples of 3 left in the deck (6 and 9). So, the probability of drawing a multiple of 3 given that Grok has already drawn an even number is 2/6.
Therefore, the probability of Grok drawing an even number, then a multiple of 3 is 3/7 * 2/6 = 1/7.
The probability that the first card Grok draws has an even number is 3/4. There are 3 even numbered cards (2, 4, and 6) and 7 total cards, so the probability of drawing an even number is 3/7.
The probability that the second card Grok draws has a multiple of 3 is 2/4. There are 2 multiples of 3 (3 and 6) and 4 total cards remaining, so the probability of drawing a multiple of 3 is 2/4.
The probability that both events happen is the product of the probabilities of each event happening. In this case, the probability is 3/4 * 2/4 = 3/8.
Therefore, the probability that Professor Grok draws two cards from Ms. Q's deck at random without replacement, where the first card has an even number and the second card has a multiple of 3 is 3/8.
