+82 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.
1 2 3 4 5 6 7
1 2 3 4 5 6 7
1 2 3 4 5 6 7
1 2 3 4 5 6 7
A) Yunseol draws five cards from Ms. Q's deck. What is the probability that exactly two cards have the same number?
B) 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 an odd number?
C) 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
+1537 (A) To find the probability that exactly two cards have the same number, we can use the following steps:
Choose the two cards that have the same number. There are 7 ways to do this.
Choose the remaining three cards. There are 24 ways to do this.
Multiply the number of ways to choose the two cards with the same number by the number of ways to choose the remaining three cards. This gives us 7⋅24=168ways to draw five cards such that exactly two cards have the same number.
Another solution
We can also use the following formula to calculate the probability of drawing exactly two cards with the same number:
P(exactly two cards with the same number) = \frac{7 \cdot \binom{4}{2} \cdot \binom{3}{3}}{\binom{28}{5}}
where:
7 is the number of ways to choose the two cards that have the same number
(24)is the number of ways to choose the remaining two cards from the same color as the two cards with the same number
(33) is the number of ways to choose the remaining card from a different color than the two cards with the same number
(528)is the total number of ways to draw five cards from the deck
Plugging in the values, we get:
P(exactly two cards with the same number) = \frac{7 \cdot 6 \cdot 1}{24192} = \boxed{\frac{42}{72576}}
This is the same probability as we calculated before.
+1537 (B)
To find the probability that the first card Professor Grok draws has an even number, and the second card Professor Grok draws has an odd number, we can use the following steps:
Choose an even number for the first card. There are 4 ways to do this.
Choose an odd number for the second card. There are 3 ways to do this.
Multiply the number of ways to choose the first card by the number of ways to choose the second card. This gives us 4⋅3=12 ways to draw two cards such that the first card has an even number, and the second card has an odd number.
Alternatively, we can use the following formula:
Probability = (Number of favorable outcomes) / (Total number of outcomes)
In this case, the number of favorable outcomes is the number of ways to draw two cards such that the first card has an even number and the second card has an odd number. There are 4 ways to choose an even number for the first card, and 3 ways to choose an odd number for the second card, so there are 4⋅3=12 favorable outcomes.
The total number of outcomes is the number of ways to draw two cards from a deck of 28 cards. There are (228)=24192ways to do this.
Therefore, the probability that Professor Grok draws two cards such that the first card has an even number and the second card has an odd number is:
Probability = 12 / 24192 = 3/5871
This is the same probability as we calculated before.
