Cards numbered 1 to 1000 were put in a box. Ali selects a card at random.
What is the probability that Ali selects a card containing at least one '3'?

To solve this problem, we need to determine the probability that Ali selects a card containing at least one '3' from a set of cards numbered from 1 to 1000.

### Step 1: Calculate the Total Number of Cards

The total number of cards is 1000, corresponding to the numbers from 1 to 1000.

### Step 2: Find the Number of Cards Without a '3'

We need to count how many of these numbers do not contain the digit '3' at all. 

To do this, we consider each digit position (hundreds, tens, and units) and count the number of possible digits that are not '3'.

- **Hundreds place**: The hundreds digit can be 0, 1, 2, 4, 5, 6, 7, 8, or 9 (9 options).
- **Tens place**: The tens digit can also be 0, 1, 2, 4, 5, 6, 7, 8, or 9 (9 options).
- **Units place**: The units digit can also be 0, 1, 2, 4, 5, 6, 7, 8, or 9 (9 options).

So, for each position, we have 9 possible digits that are not '3'.

Thus, the total number of three-digit numbers without a '3' is:

\[
9 \times 9 \times 9 = 729
\]

### Step 3: Calculate the Number of Favorable Outcomes

Now, we need to find the number of cards that contain at least one '3'. This is the complement of the number of cards that do not contain any '3'.

The total number of cards is 1000. Therefore, the number of cards without a '3' is 729. The number of cards that contain at least one '3' is:

\[
1000 - 729 = 271
\]

### Step 4: Calculate the Probability

The probability that Ali selects a card containing at least one '3' is the number of favorable outcomes divided by the total number of outcomes:

\[
\text{Probability} = \frac{\text{Number of cards containing at least one '3'}}{\text{Total number of cards}} = \frac{271}{1000}
\]

### Final Answer

The probability that Ali selects a card containing at least one '3' is:

\[
\boxed{\frac{271}{1000} \approx 0.271}
\]

### Would you like more details or have any questions?

Here are some related questions that might interest you:

1. What is the probability that a randomly selected card contains exactly one '3'?
2. How many cards contain at least two '3's?
3. What is the probability that a randomly selected card contains no '3' and no '7'?
4. How does the probability change if the number of cards is increased to 10,000?
5. What is the probability of selecting a card that contains the digit '1' at least once?

**Tip:** When solving probability problems, always consider both the direct and complement approaches to find the number of favorable outcomes!

Cards numbered 1 to 1000 were put in a box. Ali selects a card at random. What is the probability that Ali selects a card containing at least one '3'?

Discover how to calculate the probability of selecting a card containing at least one '3' from a set of cards numbered from 1 to 1000. This problem uses concepts of probability and combinatorics, with a final answer of 0.271. Ideal for students in Grades 8-10.

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