Question 10

Explanation

The sample space ($S$) is the selection of 3 numbers from the set of whole numbers less than $10$, which is $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. The number of elements in this set is $n = 10$.

The number of ways to choose 3 numbers from 10 is:

Let's check each statement.

Statement 1

Set of even numbers: $\{0, 2, 4, 6, 8\}$ (5 numbers). Number of ways to choose 3 even numbers from 5:

Probability:

Is $\frac{1}{12} > \frac{1}{20}$?

Since $\frac{5}{60} > \frac{3}{60}$, statement 1 is True.

Statement 2

The number $3$ must be selected. So we just need to choose 2 more numbers from the remaining 9.

Probability:

Statement 2 is True.

Statement 3

Factors of 9 in the set are $\{1, 3, 9\}$ (3 numbers). Numbers that are not factors of 9 are $\{0, 2, 4, 5, 6, 7, 8\}$ (7 numbers).

The probability that at least one factor of 9 is selected is $1$ minus the probability that no factor of 9 is selected.

Number of ways to choose 3 numbers that are not factors of 9:

Probability of no factor of 9:

Probability of at least one:

Is $\frac{17}{24} < \frac{1}{2}$? $\frac{1}{2} = \frac{12}{24}$. Since $\frac{17}{24} > \frac{12}{24}$, statement 3 is False.

Statement 4

Odd prime numbers in the set: $\{3, 5, 7\}$ (3 numbers). Numbers that are not odd primes: $\{0, 1, 2, 4, 6, 8, 9\}$ (7 numbers).

The event "the selected numbers are not odd prime numbers" means all three numbers are chosen from the set of non-odd-prime numbers.

Number of ways to choose 3 numbers from 7 non-odd-prime numbers:

Probability:

Is $\frac{7}{24} < \frac{1}{3}$? $\frac{1}{3} = \frac{8}{24}$. Since $\frac{7}{24} < \frac{8}{24}$, statement 4 is True.

Conclusion

The correct statements are 1, 2, and 4. There are 3 true statements.