Question

A number x is randomly chosen from the set of natural numbers less than or equal to 100. Then the probability of the event that the chosen number satisfies the inequality $\frac{(x-15)(x-70)}{x-30}\ge0$ , is

• A. 0.36
• B. 0.47
• C. 0.48
• D. 0.49
• E. 0.46

Hint:

Math Tip: When counting integers in an inclusive range $[A, B]$, always use the formula $B - A + 1$. Forgetting the $+1$ is the most common cause of "off-by-one" errors in probability questions.

Answer 1

Answer: (E)

Concept:
Probability and Algebra - Wavy Curve Method for Inequalities.
Step 1: Identify the sample space.
The number $x$ is chosen from natural numbers up to 100. $$ S = \{1, 2, 3, \dots, 100\} $$ The total number of possible outcomes is $n(S) = 100$.
Step 2: Find the critical points for the inequality.
Given inequality: $\frac{(x-15)(x-70)}{x-30} \ge 0$. Set the numerator and denominator to zero to find the critical points:

• Numerator roots: $x = 15$ and $x = 70$ (Closed circles, as they satisfy the $\ge 0$ condition).
• Denominator root: $x = 30$ (Open circle, as division by zero is undefined).

Step 3: Apply the Wavy Curve Method.
Plot the critical points on a number line: $15, 30, 70$. Check the sign of the expression in each interval:

• For $x>70$: expression is $(+)(+)/(+) \implies$ Positive
• For $30<x<70$: expression is $(+)(-)/(+) \implies$ Negative
• For $15<x<30$: expression is $(+)(-)/(-) \implies$ Positive
• For $x<15$: expression is $(-)(-)/(-) \implies$ Negative

Step 4: Determine the valid intervals.
We need the intervals where the expression is greater than or equal to zero (Positive). $$ x \in [15, 30) \cup [70, \infty) $$
Step 5: Count the favorable outcomes.
Find how many natural numbers from the sample space fall into these intervals:

• In $[15, 30)$: The integers are $15, 16, \dots, 29$. Total numbers = $29 - 15 + 1 = 15$.
• In $[70, \infty)$ up to $100$: The integers are $70, 71, \dots, 100$. Total numbers = $100 - 70 + 1 = 31$.

Total favorable outcomes $n(E) = 15 + 31 = 46$.
Step 6: Calculate the final probability.
$$ P(E) = \frac{n(E)}{n(S)} = \frac{46}{100} = 0.46 $$