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

To find the total number of teachers who suffer from depression, sum the values across all exercise levels:

$n(depression) = 9 + 2 + 1 = 12$

Now, the probability is:

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

Using conditional probability, we calculate:

$P(stress | low \ exercise) = \frac{P(stress \cap low \ exercise)}{P(low \ exercise)}$ Where:

• $P(stress \cap low \ exercise) = 38$ (teachers with stress and low exercise level).
• Total teachers with low exercise level = 50.

Thus:

$P(stress | low \ exercise) = \frac{38}{50} = 0.76$

To determine why these events are not mutually exclusive, we observe that the events can occur simultaneously. For example, there are 14 teachers with back trouble who also suffer from stress. Therefore, these two events can coexist without any conflict, as seen in the counts from the survey.