Question

A dentist was making note of his upcoming appointments with different aged patients and the reasons for their visits.
Patients under 18 Patients 19-60
Regular cleaning 6 2
Broken tooth 3 4
What is the probability that a randomly selected appointment is for a regular cleaning and is with a patient under 18?
Simplify any fractions.

Answer 1

To find the probability that a randomly selected appointment is for a regular cleaning with a patient under 18, we first need to determine the total number of appointments and the specific appointment we are interested in.

### Step 1: Determine the total number of appointments.
- Patients under 18:
- Regular cleaning: 6
- Broken tooth: 3
- Total: \(6 + 3 = 9\)

- Patients aged 19-60:
- Regular cleaning: 2
- Broken tooth: 4
- Total: \(2 + 4 = 6\)

Now, add the total for both age groups:
\[
\text{Total appointments} = 9 + 6 = 15
\]

### Step 2: Determine the number of favorable outcomes.
We are interested in the number of appointments for regular cleaning with patients under 18, which is 6.

### Step 3: Calculate the probability.
The probability \( P \) is given by the number of favorable outcomes divided by the total number of outcomes:
\[
P(\text{Regular cleaning and under 18}) = \frac{\text{Number of regular cleaning appointments for patients under 18}}{\text{Total number of appointments}} = \frac{6}{15}
\]

### Step 4: Simplify the fraction.
To simplify \(\frac{6}{15}\), we find the greatest common divisor (GCD) of 6 and 15, which is 3:
\[
\frac{6 \div 3}{15 \div 3} = \frac{2}{5}
\]

### Conclusion:
The probability that a randomly selected appointment is for a regular cleaning and is with a patient under 18 is:

\[
\boxed{\frac{2}{5}}
\]