Question

A number $ n $ is chosen at random from $ s = \{1, 2, 3, ..., 50\} $. Let $ A = \{n \in s : n \text{ is a square} \} $, $ B = \{n \in s : n \text{ is a prime}\} $, and $ C = \{n \in s : n \text{ is a square} \} $. Then, correct order of their probabilities is

• A. \( p(A)<p(B)<p(C) \)
• B. \( p(A)>p(B)>p(C) \)
• C. \( p(B)<p(A)<p(C) \)
• D. \( p(A)>p(C)>p(B) \)

Hint:

When calculating probabilities of sets, count the number of elements in each set and divide by the total number of possibilities (in this case, 50). Compare the probabilities to determine the correct order.

Answer 1

The Correct Option is B

Given the universal set \( S = \{1, 2, 3, \ldots, 50\} \), we analyze three subsets and their probabilities:

1. Set A Analysis:
Define \( A = \{ n \in S : n^2 - 27n + 50 > 0 \} \).
Factorizing the quadratic:
\[ (n - 25)(n - 2) > 0 \Rightarrow n < 2 \text{ or } n > 25 \]
Thus:
\[ A = \{1\} \cup \{26, 27, \ldots, 50\} \]
Count: \( n(A) = 1 + 25 = 26 \)
Probability: \( p(A) = \frac{26}{50} \)

2. Set B Analysis:
Define \( B = \{ n \in S : n \text{ is prime} \} \).
Primes in S:
\[ \{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47\} \]
Count: \( n(B) = 15 \)
Probability: \( p(B) = \frac{15}{50} \)

3. Set C Analysis:
Define \( C = \{ n \in S : n \text{ is a perfect square} \} \).
Squares in S:
\[ \{1, 4, 9, 16, 25, 36, 49\} \]
Count: \( n(C) = 7 \)
Probability: \( p(C) = \frac{7}{50} \)

4. Probability Comparison:
\[ \frac{26}{50} > \frac{15}{50} > \frac{7}{50} \Rightarrow p(A) > p(B) > p(C) \]

Final Answer:
The probability relationship is \(p(A) > p(B) > p(C)\).

Answer 2

The total number of elements in set \( s \) is 50. To find the probability of each set, we need to determine the number of elements in each set and divide by 50. 
- Set \( A \) consists of square numbers between 1 and 50. The squares of integers from 1 to 7 (i.e., \( 1^2, 2^2, 3^2, \dots, 7^2 \)) are in this set, so there are 7 elements in \( A \). 
- Set \( B \) consists of prime numbers between 1 and 50. The primes between 1 and 50 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47, so there are 15 elements in \( B \). 
- Set \( C \) consists of square numbers between 1 and 50, similar to set \( A \). 
Therefore, set \( C \) also has 7 elements. Now, we calculate the probabilities: \[ p(A) = \frac{7}{50}, \quad p(B) = \frac{15}{50}, \quad p(C) = \frac{7}{50}. \] Thus, the correct order of their probabilities is: \[ p(A)>p(B)>p(C). \]