Question:
The probability for a randomly selected number out of 1, 2, 3, 4, ..., 25 to be a composite number is :
The probability for a randomly selected number out of 1, 2, 3, 4, ..., 25 to be a composite number is :
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Do not simplify the fraction to \(\frac{3}{5}\) until you check the options, as multiple-choice questions often keep the total count of 25 in the denominator to make verification straightforward!
Always remember to exclude the number 1 from both prime and composite counts.
Always remember to exclude the number 1 from both prime and composite counts.
Updated On: Jul 7, 2026
- \(\frac{15}{25}\)
- \(\frac{10}{25}\)
- \(\frac{11}{25}\)
- \(\frac{9}{25}\)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Question:
We are selecting a number at random from the set of integers from 1 to 25. We need to calculate the probability that the selected number is a composite number.
Step 2: Key Formula or Approach:
1. The total number of outcomes is \(N = 25\).
2. Identify which numbers in the set \(\{1, 2, 3, \dots, 25\}\) are composite.
3. Use the classical probability formula:
\[ P(\text{Composite}) = \frac{\text{Number of composite numbers}}{\text{Total numbers}} \]
Step 3: Detailed Explanation:
1. Let the sample space be \(S = \{1, 2, 3, \dots, 25\}\). The total number of outcomes is:
\[ n(S) = 25 \]
2. Categorize the numbers in the sample space:
-
Primes: These are numbers greater than 1 with only two divisors. The prime numbers up to 25 are:
\[ \{2, 3, 5, 7, 11, 13, 17, 19, 23\} \]
Counting these, there are 9 prime numbers.
-
The number 1: As established, 1 is uniquely classified as a unit and is neither prime nor composite.
-
Composite numbers: These are the remaining natural numbers. We can find the count of composite numbers by subtracting the primes and the number 1 from the total count:
\[ \text{Count of Composites} = \text{Total numbers} - \text{Prime numbers} - 1 \]
\[ \text{Count of Composites} = 25 - 9 - 1 = 15 \]
3. List the 15 composite numbers to verify:
\[ \{4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25\} \]
This list indeed contains 15 numbers.
4. Calculate the probability of selecting a composite number:
\[ P(\text{Composite}) = \frac{15}{25} \]
Step 4: Final Answer:
The probability of selecting a composite number is \(\frac{15}{25}\), which corresponds to option (A).
We are selecting a number at random from the set of integers from 1 to 25. We need to calculate the probability that the selected number is a composite number.
Step 2: Key Formula or Approach:
1. The total number of outcomes is \(N = 25\).
2. Identify which numbers in the set \(\{1, 2, 3, \dots, 25\}\) are composite.
3. Use the classical probability formula:
\[ P(\text{Composite}) = \frac{\text{Number of composite numbers}}{\text{Total numbers}} \]
Step 3: Detailed Explanation:
1. Let the sample space be \(S = \{1, 2, 3, \dots, 25\}\). The total number of outcomes is:
\[ n(S) = 25 \]
2. Categorize the numbers in the sample space:
-
Primes: These are numbers greater than 1 with only two divisors. The prime numbers up to 25 are:
\[ \{2, 3, 5, 7, 11, 13, 17, 19, 23\} \]
Counting these, there are 9 prime numbers.
-
The number 1: As established, 1 is uniquely classified as a unit and is neither prime nor composite.
-
Composite numbers: These are the remaining natural numbers. We can find the count of composite numbers by subtracting the primes and the number 1 from the total count:
\[ \text{Count of Composites} = \text{Total numbers} - \text{Prime numbers} - 1 \]
\[ \text{Count of Composites} = 25 - 9 - 1 = 15 \]
3. List the 15 composite numbers to verify:
\[ \{4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25\} \]
This list indeed contains 15 numbers.
4. Calculate the probability of selecting a composite number:
\[ P(\text{Composite}) = \frac{15}{25} \]
Step 4: Final Answer:
The probability of selecting a composite number is \(\frac{15}{25}\), which corresponds to option (A).
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