Question

select the correct answer.
a number is selected at random from the set {2, 3, 4, ..., 10}. which event, by definition, covers the entire sample space of this experiment?
o a. the number is a positive number that is neither prime nor composite.
o b. the number is a positive number that is even or less than 11.
o c. the number is a positive number not divisible by 5.
o d. the number is a positive number greater than 2.

Explanation:

Step1: Analyze Option A

Prime numbers have only two distinct positive divisors: \(1\) and itself. Composite numbers have more than two positive divisors. In the set \(\{2,3,4,\cdots,10\}\), every number is either prime (e.g., \(2\), \(3\), \(5\), \(7\)) or composite (e.g., \(4 = 2\times2\), \(6=2\times3\), \(8 = 2\times4\), \(9=3\times3\), \(10 = 2\times5\)). There is no number in this set that is neither prime nor composite.

Step2: Analyze Option B

The set is \(\{2,3,4,\cdots,10\}\). All numbers in this set are positive numbers. Also, \(2<11\), \(3<11\), \(4<11\), \(5<11\), \(6<11\), \(7<11\), \(8<11\), \(9<11\), \(10<11\). And the set contains both even (\(2\), \(4\), \(6\), \(8\), \(10\)) and non - even (odd: \(3\), \(5\), \(7\), \(9\)) numbers. So, the statement "The number is a positive number that is even or less than \(11\)" covers all elements of the set \(\{2,3,4,\cdots,10\}\).

Step3: Analyze Option C

Numbers divisible by \(5\) in the set \(\{2,3,4,\cdots,10\}\) are \(5\) and \(10\). So, the statement "The number is a positive number not divisible by \(5\)" does not cover all elements of the set (since it excludes \(5\) and \(10\)).

Step4: Analyze Option D

Numbers greater than \(2\) in the set \(\{2,3,4,\cdots,10\}\) exclude \(2\). So, the statement "The number is a positive number greater than \(2\)" does not cover all elements of the set (since it excludes \(2\)).

Answer:

B. The number is a positive number that is even or less than 11.