a dentist was making note of his upcoming appointments with different aged patients and the reasons for…

Explanation:

Step1: Find total broken - tooth appointments

The number of broken - tooth appointments for patients under 18 is 2, and for patients 19 - 60 is 3. So the total number of broken - tooth appointments is $2 + 3=5$.

Step2: Find broken - tooth appointments for patients 19 - 60

The number of broken - tooth appointments for patients 19 - 60 is 3.

Step3: Calculate the conditional probability

The formula for conditional probability $P(A|B)=\frac{P(A\cap B)}{P(B)}$. In terms of counts, if $A$ is the event of having a patient 19 - 60 and $B$ is the event of having a broken - tooth appointment, then the probability that a randomly selected appointment is with a patient 19 - 60 given that the appointment is for a broken tooth is $\frac{\text{Number of 19 - 60 and broken - tooth appointments}}{\text{Total number of broken - tooth appointments}}=\frac{3}{5}$.

Answer:

$\frac{3}{5}$