What is the probability that a given two-digit number is divisible by 15?

Answer

To find the probability that a given two-digit number is divisible by 15, we can follow these steps:

### Step 1: Identify the Sample Space
The two-digit numbers range from 10 to 99. We need to count how many two-digit numbers there are.

- The smallest two-digit number is 10.
- The largest two-digit number is 99.

To find the total count of two-digit numbers:
\[ \text{Total two-digit numbers} = 99 - 10 + 1 = 90 \]

### Step 2: Identify the Event of Interest
Next, we need to find the two-digit numbers that are divisible by 15. A number is divisible by 15 if it is divisible by both 3 and 5.

The two-digit multiples of 15 can be found by listing them:
- The first two-digit multiple of 15 is 15.
- The last two-digit multiple of 15 is 90.

Now, we can list the two-digit multiples of 15:
- 15, 30, 45, 60, 75, 90

### Step 3: Count the Favorable Outcomes
Now we count the numbers we found in the previous step:
- The two-digit numbers divisible by 15 are: 15, 30, 45, 60, 75, 90.
- There are a total of 6 such numbers.

### Step 4: Calculate the Probability
The probability \( P \) of an event is given by the formula:
\[ P(E) = \frac{n(E)}{n(S)} \]
where:
- \( n(E) \) is the number of favorable outcomes (numbers divisible by 15),
- \( n(S) \) is the total number of outcomes (total two-digit numbers).

Substituting the values we found:
\[ P(E) = \frac{6}{90} \]

### Step 5: Simplify the Probability
Now we simplify the fraction:
\[ P(E) = \frac{6 \div 6}{90 \div 6} = \frac{1}{15} \]

### Final Answer
Thus, the probability that a given two-digit number is divisible by 15 is:
\[ \frac{1}{15} \]

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