Question

In a certain show, a lottery ticket is numbered consecutively from 100 through 999 (both inclusive). What is the probability that a randomly selected ticket will have a number with a ten's digit as "3"?

• A. 1/5
• B. 90/899
• C. 1/10
• D. 1/11
• E. 10/111

Hint:

For counting problems involving digits, it's often easiest to think about the number of choices for each digit position (hundreds, tens, units) and then multiply them. Also, a quick way to think about this specific problem is that for any given hundreds digit, there are 10 numbers with a '3' in the tens place (e.g., 130-139). Since there are 9 possible hundreds digits (1-9), the total is \(9 \times 10 = 90\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a probability problem. The probability is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.
Step 2: Detailed Explanation:
1. Calculate the total number of possible outcomes.
The tickets are numbered from 100 to 999, inclusive. The total number of tickets is given by: \[ \text{Total Numbers} = (\text{Last Number} - \text{First Number}) + 1 \] \[ \text{Total Numbers} = (999 - 100) + 1 = 899 + 1 = 900 \] So, there are 900 possible outcomes.
2. Calculate the number of favorable outcomes.
We need to find the number of tickets that have a "3" in the ten's digit. These are numbers of the form H3T, where H is the hundreds digit and T is the units digit.

The hundreds digit (H) can be any digit from 1 to 9 (9 choices).
The ten's digit is fixed as 3 (1 choice).
The units digit (T) can be any digit from 0 to 9 (10 choices).
The total number of such numbers is the product of the number of choices for each digit: \[ \text{Favorable Outcomes} = 9 \times 1 \times 10 = 90 \] So, there are 90 numbers between 100 and 999 that have a 3 in the ten's digit.
3. Calculate the probability.
\[ P(\text{ten's digit is 3}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \] \[ P = \frac{90}{900} = \frac{1}{10} \] Step 3: Final Answer:
The probability is 1/10. This corresponds to option (C).