To solve the problem of finding the probability that a randomly drawn ticket from a bag containing tickets numbered 1 to 20 is a multiple of 2 or 5, we can follow these steps:
### Step 1: Define the Sample Space
The sample space (S) consists of all the tickets, which are numbered from 1 to 20. Therefore, the total number of outcomes in the sample space is:
\[ |S| = 20 \]
**Hint:** The sample space includes all possible outcomes of the experiment.
### Step 2: Identify the Events
Let:
- Event A be the event that the number drawn is a multiple of 2.
- Event B be the event that the number drawn is a multiple of 5.
### Step 3: List the Elements of Each Event
- **Multiples of 2 (Event A):** The multiples of 2 from 1 to 20 are:
\[ A = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\} \]
Thus, \( |A| = 10 \).
- **Multiples of 5 (Event B):** The multiples of 5 from 1 to 20 are:
\[ B = \{5, 10, 15, 20\} \]
Thus, \( |B| = 4 \).
**Hint:** Count the numbers that satisfy the conditions for each event.
### Step 4: Find the Intersection of the Events
The intersection of events A and B (i.e., numbers that are multiples of both 2 and 5) consists of the common elements:
\[ A \cap B = \{10, 20\} \]
Thus, \( |A \cap B| = 2 \).
**Hint:** Look for numbers that are common to both lists of multiples.
### Step 5: Use the Probability Formula
To find the probability of the union of events A and B (i.e., the probability that a ticket drawn is a multiple of 2 or 5), we use the formula:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]
### Step 6: Calculate the Individual Probabilities
- The probability of event A:
\[
P(A) = \frac{|A|}{|S|} = \frac{10}{20} = \frac{1}{2}
\]
- The probability of event B:
\[
P(B) = \frac{|B|}{|S|} = \frac{4}{20} = \frac{1}{5}
\]
- The probability of the intersection of A and B:
\[
P(A \cap B) = \frac{|A \cap B|}{|S|} = \frac{2}{20} = \frac{1}{10}
\]
### Step 7: Substitute into the Formula
Now, substituting the values into the formula:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]
\[
P(A \cup B) = \frac{1}{2} + \frac{1}{5} - \frac{1}{10}
\]
### Step 8: Find a Common Denominator and Calculate
The common denominator for 2, 5, and 10 is 10. Converting each fraction:
\[
P(A \cup B) = \frac{5}{10} + \frac{2}{10} - \frac{1}{10} = \frac{5 + 2 - 1}{10} = \frac{6}{10} = \frac{3}{5}
\]
### Final Answer
Thus, the probability that the ticket drawn is a multiple of 2 or 5 is:
\[
\boxed{\frac{3}{5}}
\]
