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# Probability

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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.

Yunseol draws 5 cards from Ms. Q's deck. What is the probability that three cards have the same number?

Mar 5, 2023

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There are a total of 4 different colors and 7 different numbers, so there are 4 x 7 = 28 cards in total.

To find the probability that Yunseol draws 3 cards with the same number, we can break it down into cases based on the number that appears on the three cards.

Case 1: Yunseol draws 3 cards with the number 1.
There are 4 colors to choose from for the first card, 3 colors left to choose from for the second card, and 2 colors left to choose from for the third card. So there are 4 x 3 x 2 = 24 ways to choose the colors for the cards. Once the colors are chosen, there is only 1 card with the number 1 of each color, so there is only 1 way to choose the cards for each color. Therefore, there are 1 x 1 x 1 = 1 way to choose the cards for each set of colors. Finally, there are 7 ways to choose which number appears on the three cards. So there are a total of 24 x 7 = 168 ways to draw 3 cards with the number 1.

Case 2: Yunseol draws 3 cards with the number 2.
Using similar reasoning as above, there are 24 ways to choose the colors for the cards, 1 way to choose the cards for each color, and 7 ways to choose which number appears on the three cards. So there are a total of 24 x 7 = 168 ways to draw 3 cards with the number 2.

Cases 3-6: Yunseol draws 3 cards with the number 3, 4, 5, or 6.
By symmetry, the number of ways to draw 3 cards with each of these numbers is the same as the number of ways to draw 3 cards with the number 2. So there are a total of 4 x 168 = 672 ways to draw 3 cards with the same number.

To choose the remaining 2 cards, there are 25 cards left in the deck (since Yunseol has already drawn 3 cards). The probability of drawing a card with a different number than the first 3 cards is 20/25, or 4/5, for each draw. So the probability of drawing 2 cards with a different number than the first 3 cards is (4/5)^2 = 16/25.

Therefore, the probability of drawing 3 cards with the same number and then 2 cards with different numbers is (672/28 C 3) x 16/25 = 0.384 or approximately 38.4%.

So there is a 38.4% chance that Yunseol draws 3 cards with the same number.

Mar 5, 2023