The Venn diagram illustrates the specialized competencies of 40 researchers in a data science lab. The events are defined as:
PPP: Researcher is proficient in Python
SSS: Researcher is proficient in SQL
BBB: Researcher is proficient in Bioinformatics
Within the universal set of 40 researchers:

Identify two of these events that are mutually exclusive. Justify your answer.
Determine whether or not the events PPP and SSS are independent.
One researcher is selected at random. Calculate the probability that this person:
is proficient in Bioinformatics,
is proficient in both Python and SQL.
Given that this researcher is proficient in Python, find the probability that they are also proficient in SQL.
200 exam-style questions on OCR (MEI) A Level Maths 2.3 Probability, covering 2.3.1 Calculate the probability of an event, 2.3.2 Complementary events, 2.3.3 Expected frequency of an event, 2.3.4 Diagrams to calculate probabilities, 2.3.5 Mutually exclusive and independent events, 2.3.6 Add probabilities for mutually exclusive events, 2.3.7 Multiply probabilities for independent events, 2.3.8 Mutually exclusive and independent events (notation) (A-level only), 2.3.9 Venn diagrams for probabilities (A-level only), 2.3.10 Conditional probabilities (A-level only), and 2.3.11 Independence via conditional probability (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.