Question

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?

Answer 1

The probability that the first card drawn is red or green, and the second card drawn is odd and yellow is 1/18. This is calculated by first finding the probability of drawing a red or green card and then multiplying it by the probability of drawing an odd yellow card second, assuming the first draw didn't affect the second.

Step-by-step explanation:

To solve this problem, we need to calculate the combined probability of two independent events happening in sequence. Since the drawing of cards is without replacement, we treat the events as dependent. Let's represent 'the first card is red or green' by R/G and 'the second card is odd and yellow' by OY.

Focusing on the first event R/G, there are 2 possible colors out of 4 and 7 different numbers. So the probability for R/G is:

P(R/G) = (Number of favorable outcomes for R/G) / (Total number of outcomes) = (2 colors × 7 numbers) / (4 colors × 7 numbers) = 14 / 28 = 1/2.

Now for the second event OY, considering one card is already taken, there are now 27 cards left. There are 3 odd yellow cards initially, but the first draw might affect this. Assuming R/G didn't remove an odd yellow card (which has a higher chance than removing one), the probability of drawing an odd yellow card is:

P(OY) = 3 / 27

Since these events are dependent, we multiply the probabilities:

Combined probability P(R/G AND OY) = P(R/G) × P(OY) = (1/2) × (3/27) = 3/54 = 1/18

Therefore, the probability that the first card drawn is red or green, and the second card drawn is odd and yellow is 1/18.

Answer 2

The probability that the first card drawn is red or green, and the second card drawn is odd and yellow is approximately 0.0185 or 1/54.

To find the probability that the first card drawn is red or green and the second card drawn is odd and yellow, we can break down the problem into two parts:

Part 1: Probability of drawing a red or green card first.

There are four colors in the deck: red, green, blue, and yellow. Among these, we want to find the probability of drawing a red or green card first.

Step 1: Determine the total number of red or green cards.

Since each color has one card for each number, there are 7 red cards and 7 green cards in the deck. So, the total number of red or green cards is 7 + 7 = 14.

Step 2: Determine the total number of cards in the deck.

Since there are 4 colors and 7 numbers, the total number of cards in the deck is 4 * 7 = 28.

Step 3: Calculate the probability.

The probability of drawing a red or green card first can be calculated by dividing the total number of red or green cards by the total number of cards in the deck:

Probability = (Number of red or green cards) / (Total number of cards)

= 14 / 28

= 1/2

= 0.5

Part 2: Probability of drawing an odd and yellow card second.

Since the first card drawn is red or green, we are left with 27 cards in the deck, including 6 yellow cards. Among these, we want to find the probability of drawing an odd and yellow card second.

Step 1: Determine the total number of odd and yellow cards.

Since each color has one card for each number, there is one odd and yellow card in the deck, which is the card labeled with number 3.

Step 2: Determine the total number of remaining cards in the deck.

After drawing the first card, we are left with 27 cards in the deck.

Step 3: Calculate the probability.

The probability of drawing an odd and yellow card second can be calculated by dividing the total number of odd and yellow cards by the total number of remaining cards in the deck:

Probability = (Number of odd and yellow cards) / (Total number of remaining cards)

= 1 / 27

Now, to find the probability of both events occurring together, we multiply the probabilities of each event:

Probability of both events = Probability of drawing a red or green card first * Probability of drawing an odd and yellow card second

= 0.5 * 1/27

= 1/54

≈ 0.0185

Hence, the probability that the first card drawn is red or green, and the second card drawn is odd and yellow is approximately 0.0185 or 1/54.