Question

Find the probability that the age of the selected student is a composite number.

Answer 1

Question:
The ages (in years) of 10 students in a group are:
14, 15, 15, 16, 16, 16, 17, 17, 19, 19.
If a student is selected at random, find the probability that the student’s age is a composite number.

Answer 2

X can be any age in the data:
X ∈{14, 15, 16, 17, 19}

Answer 3

Step 1: Prepare the frequency distribution table:
Ages and their frequencies:
\[ \begin{aligned} &\text{Age 14: } f = 1 \\ &\text{Age 15: } f = 2 \\ &\text{Age 16: } f = 3 \\ &\text{Age 17: } f = 2 \\ &\text{Age 19: } f = 2 \\ \end{aligned} \]
Step 2: Use the formula for mean:
\[ \mu = \sum x_i P(X = x_i) = \sum \frac{x_i f_i}{N} \] where \( N = 10 \) (total frequency).
Compute: \[ \mu = \frac{1 \cdot 14 + 2 \cdot 15 + 3 \cdot 16 + 2 \cdot 17 + 2 \cdot 19}{10} \] \[ = \frac{14 + 30 + 48 + 34 + 38}{10} = \frac{164}{10} = 16.4 \] Final Answer:
Mean age = 16.4 years

Answer 4

Step 1: Extract ages greater than 15:
\[ \text{Ages }>15 = \{16, 17, 17, 16, 19, 16, 19\} \] Total count = 7 students.
Step 2: Identify which of these are prime numbers:
Prime ages in the list = 17, 19
17 occurs 2 times, 19 occurs 2 times ⇒ Total prime occurrences = 4
Step 3: Use conditional probability:
\[ P(\text{prime} \mid \text{age}>15) = \frac{\text{Number of prime ages>15}}{\text{Total number of ages>15}} = \frac{4}{7} \] Final Answer:
\( \boxed{\dfrac{4}{7}} \)