Answer: The probability of drawing at least two cards with a number larger than 2 is 8/45 or approximately 0.1778.
Explanation:
To determine the probability of drawing at least two cards with a number larger than 2, we need to calculate the probability of two cases:
Case 1: Three cards drawn have numbers larger than 2.
The probability of drawing a card with a number larger than 2 on the first draw is 2/3 since there are 2 cards with a number larger than 2 out of 3 cards remaining in the deck. The probability of drawing a card with a number larger than 2 on the second draw, without replacement, is 2/5 since there are 2 cards with a number larger than 2 out of 5 cards remaining in the deck. The probability of drawing a card with a number larger than 2 on the third draw, without replacement, is 1/3 since there is 1 card with a number larger than 2 out of 3 cards remaining in the deck.
So, the probability of drawing three cards with numbers larger than 2 is:
(2/3) x (2/5) x (1/3) = 4/45
Case 2: Two cards drawn have numbers larger than 2.
The probability of drawing a card with a number larger than 2 on the first draw is 2/3, as before. The probability of drawing a card with a number larger than 2 on the second draw, without replacement, is 2/5, as before. However, the third card must have a number less than or equal to 2. The probability of drawing a card with a number less than or equal to 2 on the third draw, without replacement, is 4/8 since there are 4 cards with numbers less than or equal to 2 out of 8 cards remaining in the deck.
So, the probability of drawing two cards with numbers larger than 2 and one card with a number less than or equal to 2 is:
(2/3) x (2/5) x (4/8) = 4/75
Finally, we add the probabilities of the two cases to get the probability of drawing at least two cards with a number larger than 2:
4/45 + 4/75 = 8/45
Therefore, the probability of drawing at least two cards with a number larger than 2 is 8/45 or approximately 0.1778.
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Sources:
"Introduction to Probability" by Joseph K. Blitzstein and Jessica Hwang. This book provides a comprehensive introduction to probability theory, including calculations involving drawing cards.
"Statistics and Probability with Applications for Engineers and Scientists" by Bhisham C. Gupta and Irwin Guttman. This book provides an overview of probability and statistics, including examples and applications to engineering and science.
"Probability: Theory and Examples" by Rick Durrett. This book provides a detailed introduction to probability theory, including topics such as independence, conditional probability, and the law of large numbers.
"A First Course in Probability" by Sheldon Ross. This book provides a comprehensive introduction to probability theory, including examples and applications to a wide range of fields.
"Probability and Random Processes" by Geoffrey Grimmett and David Stirzaker. This book provides a thorough introduction to probability theory and random processes, including calculations involving drawing cards.