What is the probability of getting a composite number in the list of natural numbers from 1 to 50?
Calculating Probability of Composite Numbers (1 to 50)
The question asks for the probability of selecting a composite number from the list of natural numbers starting from 1 up to 50. To find this probability, we need to determine the total number of possible outcomes and the number of favorable outcomes (composite numbers) in the given range.
Understanding Probability
Probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes. Mathematically, it is expressed as:
\(\text{Probability (Event)} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}\)
Identifying the Sample Space
The natural numbers from 1 to 50 are the set of possible outcomes. This set is \(\{1, 2, 3, \dots, 50\}\).
The total number of possible outcomes is 50.
Defining Composite Numbers
A composite number is a natural number greater than 1 that has at least one divisor other than 1 and itself. In simpler terms, a composite number is a natural number greater than 1 that is not a prime number.
- The number 1 is neither prime nor composite.
- Prime numbers are natural numbers greater than 1 that have only two distinct positive divisors: 1 and themselves (e.g., 2, 3, 5, 7).
- Composite numbers are natural numbers greater than 1 that are not prime (e.g., 4, 6, 8, 9).
Identifying Composite Numbers from 1 to 50
To find the composite numbers from 1 to 50, we can exclude 1 and all the prime numbers in this range. The prime numbers from 1 to 50 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
There are 15 prime numbers between 1 and 50.
The total number of natural numbers from 1 to 50 is 50.
The number 1 is not composite.
The numbers greater than 1 in the list are $50 - 1 = 49$. These 49 numbers are either prime or composite.
Number of composite numbers = (Total numbers from 1 to 50) - (Number 1) - (Number of prime numbers from 1 to 50)
Number of composite numbers = $50 - 1 - 15 = 34$.
Alternatively, Number of composite numbers = (Total numbers > 1 up to 50) - (Number of prime numbers up to 50)
Number of composite numbers = $49 - 15 = 34$.
So, the number of favorable outcomes (composite numbers) is 34.
Calculating the Probability
Now we can calculate the probability using the probability formula:
\(\text{Probability (Getting a composite number)} = \frac{\text{Number of Composite Numbers}}{\text{Total Number of Natural Numbers from 1 to 50}}\)
\(\text{Probability} = \frac{34}{50}\)
We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.
\(\text{Probability} = \frac{34 \div 2}{50 \div 2} = \frac{17}{25}\)
The probability of getting a composite number in the list of natural numbers from 1 to 50 is \(\frac{17}{25}\).
Summary of Steps
- Identify the total number of outcomes (natural numbers from 1 to 50).
- Identify the composite numbers within this range.
- Count the number of composite numbers.
- Calculate the probability by dividing the count of composite numbers by the total number of outcomes.
- Simplify the resulting fraction.
| Description | Count |
|---|---|
| Total natural numbers (1 to 50) | 50 |
| Number 1 (neither prime nor composite) | 1 |
| Prime numbers (1 to 50) | 15 |
| Composite numbers (1 to 50) | 34 |
Probability = \(\frac{\text{Number of Composite Numbers}}{\text{Total Numbers}} = \frac{34}{50} = \frac{17}{25}\).
Revision Table: Probability and Number Types
| Term | Definition | Example (from 1 to 10) |
|---|---|---|
| Natural Number | Positive integers starting from 1. | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 |
| Prime Number | Natural number > 1 with only two divisors: 1 and itself. | 2, 3, 5, 7 |
| Composite Number | Natural number > 1 that is not prime. | 4, 6, 8, 9, 10 |
| Probability | Measure of the likelihood of an event occurring. | Ratio of favorable outcomes to total outcomes. |
Additional Information: Prime vs. Composite
Understanding the difference between prime and composite numbers is fundamental for solving this type of probability problem. The number 1 is unique because it only has one divisor (itself), so it does not fit the definition of either prime or composite. All natural numbers greater than 1 are either prime or composite. This classification helps us count the composite numbers effectively by subtracting the count of primes and the number 1 from the total count of numbers in the range.
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