One ticket is drawn from a box containing 100 tickets numbered from 1 to 100. If the number on the selected ticket is x then probability that `x+ 1/x gt 2` is

Options

A. 0
B. `99/100`
C. 1
D. `9/100`

Answer

To solve the problem, we need to find the probability that \( x + \frac{1}{x} > 2 \) when a ticket numbered \( x \) is drawn from a box containing tickets numbered from 1 to 100.

### Step-by-Step Solution:

1. **Understand the Condition**: We start with the inequality \( x + \frac{1}{x} > 2 \).

2. **Rearrange the Inequality**: We can manipulate the inequality:
\[ x + \frac{1}{x} > 2 \]
Multiply both sides by \( x \) (since \( x > 0 \)):
\[ x^2 + 1 > 2x \]

3. **Rearranging Further**: Rearranging gives us:
\[ x^2 - 2x + 1 > 0 \]
This can be factored as:
\[ (x - 1)^2 > 0 \]

4. **Analyzing the Factored Inequality**: The expression \( (x - 1)^2 > 0 \) is true for all \( x \) except when \( x = 1 \). Therefore, \( (x - 1)^2 = 0 \) at \( x = 1 \) and is positive for all other values of \( x \).

5. **Identify Valid Values of \( x \)**: Since \( x \) can take any integer value from 1 to 100, the values that satisfy \( (x - 1)^2 > 0 \) are all integers from 2 to 100. Thus, the valid values of \( x \) are:
\[ x \in \{2, 3, 4, \ldots, 100\} \]

6. **Count the Favorable Outcomes**: The total number of integers from 1 to 100 is 100. The integers that do not satisfy the condition is only \( x = 1 \). Therefore, the favorable outcomes are:
\[ 100 - 1 = 99 \]

7. **Calculate the Probability**: The probability \( P \) that \( x + \frac{1}{x} > 2 \) is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{99}{100} \]

### Final Answer:
The probability that \( x + \frac{1}{x} > 2 \) is \( \frac{99}{100} \).