Question

if a number is chosen randomly from the set of numbers {1, 2, 3, 4, ..., 13}, find the probability of getting a composite number. the probability of getting a composite number is □. (simplify your answer.)

Explanation:

Step1: Determine the total number of elements

The set is \(\{1, 2, 3,\cdots,13\}\). Using the formula for the number of elements in an arithmetic sequence \(n=a_{n}-a_{1}+1\) (where \(a_{n}\) is the last term, \(a_{1}\) is the first term), here \(a_{1} = 1\), \(a_{n}=13\), so \(n = 13\).

Step2: Identify composite numbers in the set

Composite numbers are positive integers that have at least one positive divisor other than \(1\) and itself.
The composite numbers in the set \(\{1,2,\cdots,13\}\) are \(4,6,8,9,10,12\). The number of composite numbers \(m = 6\).

Step3: Calculate the probability

The probability formula is \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
Substitute \(m = 6\) (number of composite numbers) and \(n = 13\) (total numbers) into the formula: \(P=\frac{6}{13}\).

Answer:

\(\frac{6}{13}\)