The following shows the results of a survey on the types of exercise taken by a group of 100 people - Edexcel - A-Level Maths Statistics - Question 6 - 2012 - Paper 1

To find the probability that a randomly selected person takes none of these exercises, start by calculating the total involved in exercises:

• Total involved = Total people - people not involved
• Total involved = 100 - (people who do any exercise). From the Venn Diagram: $No Exercise = 100 - (Total with Exercise)$ Assuming we find that 93 people do some type of exercise, then: $P(No Exercise) = \frac{7}{100} = 0.07$.

To find the probability of someone who swims but does not run, refer to the Venn Diagram:

• This would be the count in S but not in R, which is the number who swim (48) minus those who both swim and run (15 from step (a)). Thus: $P(S \cap \neg R) = \frac{48 - 15}{100} = \frac{33}{100} = 0.33$.

The count of people taking at least two types is those involved in the overlapping sections of the Venn Diagram:

• This includes the intersections of R and S, S and C, R and C, and those who do all three (25):
• Adding these segments: $Total = (15 + 10 + 5 + 25) = 55$. Thus, therefore the probability: $P(At \ least \ 2) = \frac{55}{100} = 0.55$.

Given that Jason runs, we need to find the probability that he swims and does not cycle: To solve:

1. Filter for those who run (65 total).
2. Determine those who swim and run but not cycle (15 based on previously calculated values from the Venn Diagram): $P(S \cap \neg C | R) = \frac{15}{65}$.
3. Thus, calculate this to yield: $P(S \cap \neg C | R) = \frac{15}{65} = \frac{3}{13}$.