One number is chosen at random from the number 1 to 100. Find the probability that it is divisible by 4 or 10.

Answer

To solve the problem of finding the probability that a number chosen at random from 1 to 100 is divisible by 4 or 10, we can follow these steps:

### Step 1: Define the Events
Let:
- Event A = the event that a number is divisible by 4.
- Event B = the event that a number is divisible by 10.

### Step 2: Determine the Sample Space
The total number of possible outcomes (sample space) when choosing a number from 1 to 100 is 100.

### Step 3: Find the Numbers Divisible by 4
To find the numbers divisible by 4, we can list them:
- The numbers divisible by 4 from 1 to 100 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100.

Counting these, we find there are 25 numbers divisible by 4.

### Step 4: Find the Numbers Divisible by 10
Next, we list the numbers divisible by 10:
- The numbers divisible by 10 from 1 to 100 are: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100.

Counting these, we find there are 10 numbers divisible by 10.

### Step 5: Find the Numbers Divisible by Both 4 and 10
Now, we need to find the numbers that are divisible by both 4 and 10 (i.e., divisible by 20):
- The numbers divisible by 20 from 1 to 100 are: 20, 40, 60, 80, 100.

Counting these, we find there are 5 numbers that are divisible by both 4 and 10.

### Step 6: Use the Formula for Probability of A Union B
To find the probability of A union B (i.e., the probability that a number is divisible by 4 or 10), we use the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Where:
- \( P(A) = \frac{\text{Number of favorable outcomes for A}}{\text{Total outcomes}} = \frac{25}{100} \)
- \( P(B) = \frac{\text{Number of favorable outcomes for B}}{\text{Total outcomes}} = \frac{10}{100} \)
- \( P(A \cap B) = \frac{\text{Number of favorable outcomes for both A and B}}{\text{Total outcomes}} = \frac{5}{100} \)

### Step 7: Calculate the Probabilities
Now substituting the values into the formula:
\[ P(A \cup B) = \frac{25}{100} + \frac{10}{100} - \frac{5}{100} \]

Calculating this gives:
\[ P(A \cup B) = \frac{25 + 10 - 5}{100} = \frac{30}{100} = \frac{3}{10} \]

### Final Answer
Thus, the probability that a number chosen at random from 1 to 100 is divisible by 4 or 10 is:
\[ \frac{3}{10} \]

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