A survey was conducted into the health of 120 teachers - AQA - A-Level Maths Mechanics - Question 14 - 2019 - Paper 3

From the table, we see that the total number of teachers suffering from depression is the sum of all exercise levels:

• Low exercise: 9
• Medium exercise: 2
• High exercise: 1

Thus, the total number suffering from depression is:

$Total \, depression = 9 + 2 + 1 = 12$

The probability of a teacher suffering from depression is:

$P(suffers \, from \, depression) = \frac{12}{120} = \frac{1}{10}$

To find the conditional probability, we will use the formula for conditional probability:

$P(A|B) = \frac{P(A \, \cap \, B)}{P(B)}$

Let A be the event of suffering from stress, and B be the event of having a low exercise level.

From the table, the number of teachers suffering from stress and having low exercise is 38. The total number of teachers with low exercise is 50.

Thus:

$P(stress|low \, exercise) = \frac{38}{50}$

To explain this, we need to understand the definition of mutually exclusive events. Two events are mutually exclusive if they cannot occur at the same time. In this survey, a teacher can have both back trouble and stress simultaneously. For example, there are teachers counted in both categories:

• Back trouble: 14 with low exercise
• Stress: 38 with low exercise

They are not mutually exclusive because it's possible for a teacher to experience both conditions at the same time. Therefore, we conclude that:

The events ‘suffers from back trouble’ and ‘suffers from stress’ are not mutually exclusive.