A single number is chosen from the digits 1 through 30. Find the probability that the number is an even number or a number divisible by 3.

We are tasked with finding the probability that a number chosen from the digits 1 through 30 is either an even number or a number divisible by 3.

### Step 1: Total number of outcomes
The total number of digits from 1 to 30 is 30, so the total number of possible outcomes is:
\[
\text{Total outcomes} = 30
\]

### Step 2: Counting favorable outcomes
We now need to count how many numbers are either:
1. Even numbers.
2. Divisible by 3.

#### Even numbers:
The even numbers between 1 and 30 are:
\[
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30
\]
There are 15 even numbers.

#### Numbers divisible by 3:
The numbers divisible by 3 between 1 and 30 are:
\[
3, 6, 9, 12, 15, 18, 21, 24, 27, 30
\]
There are 10 numbers divisible by 3.

### Step 3: Applying the inclusion-exclusion principle
Some numbers are counted twice because they are both even and divisible by 3. These numbers are:
\[
6, 12, 18, 24, 30
\]
There are 5 such numbers.

Using the inclusion-exclusion principle, the number of favorable outcomes is:
\[
\text{Favorable outcomes} = (\text{Even numbers}) + (\text{Divisible by 3}) - (\text{Both even and divisible by 3})
\]
\[
\text{Favorable outcomes} = 15 + 10 - 5 = 20
\]

### Step 4: Calculating the probability
The probability is the ratio of favorable outcomes to the total number of outcomes:
\[
P = \frac{20}{30} = \frac{2}{3}
\]

### Final Answer:
The probability that the chosen number is either an even number or divisible by 3 is \( \frac{2}{3} \).

Would you like further clarification or details?

Here are some related questions to expand your understanding:
1. What is the probability of selecting a number divisible by both 2 and 3?
2. How many numbers between 1 and 30 are divisible by 5?
3. What is the probability of selecting an odd number from 1 to 30?
4. How many prime numbers exist between 1 and 30?
5. What is the probability that a number chosen from 1 to 30 is divisible by 4?

**Tip**: When dealing with probabilities of unions (like "or" conditions), using the inclusion-exclusion principle helps avoid overcounting.

Learn how to calculate the probability of selecting a number from 1 to 30 that is either even or divisible by 3. This problem involves key concepts in probability and the inclusion-exclusion principle. Suitable for students in Grades 8-10.

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