Question

An integer is chosen from the first 100 positive integers. Probability that the chosen number is a multiple of 11 is:

• A. $\frac{1}{10}$
• B. $\frac{1}{11}$
• C. $\frac{9}{100}$
• D. $\frac{13}{100}$
• E. $\frac{11}{100}$

Hint:

Divide 100 by 11 to get 9.09; the integer part tells you how many multiples there are.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We need to find the probability of selecting a number that is a multiple of 11 from the set of the first 100 positive integers.
Step 2: Key Formula or Approach:
The probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes.
\[ P(\text{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \] Step 3: Detailed Explanation:
Total Number of Outcomes:
The experiment involves choosing an integer from the first 100 positive integers, i.e., from the set \{1, 2, 3, ..., 100\}.
The total number of possible outcomes is 100.
Number of Favorable Outcomes:
The event is choosing a number that is a multiple of 11. We need to find how many multiples of 11 are there between 1 and 100.
The multiples are 11, 22, 33, 44, 55, 66, 77, 88, 99.
The next multiple, \( 11 \times 10 = 110 \), is outside our range.
To find the number of multiples systematically, we can use integer division:
\[ \text{Number of multiples} = \left\lfloor \frac{100}{11} \right\rfloor = \lfloor 9.09... \rfloor = 9 \] So, there are 9 favorable outcomes.
Calculate the Probability:
\[ P(\text{multiple of 11}) = \frac{9}{100} \] Step 4: Final Answer:
The probability that the chosen number is a multiple of 11 is \( \frac{9}{100} \).