Example Question - prime number
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Set Theory Problem Involving Intersection of Sets in Roster Form
<p>To find \( A \cap B \) in roster form, identify the elements common to both set \( A \) and set \( B \).</p> <p>By definition from the question:</p> <p>\( A = \{x: x \leq 10, x \in \mathbb{N}, x \text{ is a prime number}\} \)</p> <p>\( B = \{x: x \in \mathbb{N}, x \text{ is even}\} \)</p> <p>First list the prime numbers less than or equal to 10 (prime numbers are numbers greater than 1 that have no positive divisors other than 1 and itself): 2, 3, 5, 7</p> <p>Now list the even natural numbers (since only 2 is both even and prime): 2</p> <p>Therefore, the intersection of set A and set B, \( A \cap B \), which includes all the elements that are both prime and even is: {2}, since 2 is the only even prime number.</p> <p>\( A \cap B = \{2\} \).</p>
Solving for Specific Types of Numbers
<p>Since the question is not conveyed in its entirety and the problem statement seems incomplete, no specific numerical solution can be provided. Further clarification on the original problem would be necessary to proceed with a valid mathematical solution.</p>
Probability of Landing on Specific Numbers on a Spinner
To solve this problem, we need to consider the two steps described: first landing on a number less than 4, and then landing on a prime number. From the image of the spinner, we can see that there are 5 sections, each containing the numbers from 1 to 5. Let's work on the first part: The probability of landing on a number less than 4. Numbers less than 4 on the spinner are 1, 2, and 3. That means there are 3 favorable outcomes out of the 5 possible outcomes. So, the probability P(less than 4) is 3/5 or 60%. Now, for the second part: The probability of landing on a prime number after we have already landed on a number less than 4. The prime numbers on the spinner are 2, 3, and 5. However, the second spin is independent of the first, meaning that we consider the probability of landing on a prime number out of all possible outcomes again - which are 2, 3, and 5 again. So, there are again 3 favorable outcomes out of the 5 possible outcomes. The probability P(prime number) is therefore also 3/5 or 60%. The overall probability of both events happening in sequence (landing on a number less than 4 and then landing on a prime number) is the product of their individual probabilities: P(less than 4) * P(prime number) = (3/5) * (3/5) = 9/25. To express this as a percentage, we compute: (9/25) * 100% = 36%. Therefore, the probability of landing on a number less than 4 and then on a prime number, when spinning this spinner twice, is 36%.
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