Using probability to assess risk Revision Notes for AQA GCSE Statistics

Using probability to assess risk

What is risk in probability?

In probability, risk refers to the likelihood that a particular event will occur based on real data collected from experiments or observations. When we want to understand the chances of an accident, injury, or other outcome happening, we calculate risk using experimental probability.

chatImportant

Fundamental Concept

Risk represents the probability of an unwanted event occurring, and this probability is determined by analysing actual occurrences in similar situations. This makes risk assessment based on real data rather than theoretical probability.

Calculating risk

The basic formula for calculating risk is:

$\text{Risk of event} = \frac{\text{Number of times event occurred}}{\text{Total number of trials}}$

This gives us a decimal value between 0 and 1, which can also be expressed as a percentage. Converting to decimals makes it much easier to compare different risks.

infoNote

When working with risk calculations, always express your final answers as decimals rather than fractions. This standardised format allows for quick and accurate comparison between different risk values.

lightbulbExample

Worked Example: Basketball Injury Risk Analysis

Using the basketball injury data shown above:

Step 1: Apply the risk formula for each injury type

Risk of finger injury = $\frac{6}{50} = 0.12$
Risk of foot injury = $\frac{8}{46} = 0.17$
Risk of knee injury = $\frac{3}{50} = 0.06$

Step 2: Compare the calculated risks From these calculations, we can see that foot injuries have the highest risk (0.17), followed by finger injuries (0.12), and knee injuries have the lowest risk (0.06).

Types of risk measurement

Absolute risk

Absolute risk is simply the probability of an event happening to any individual in a particular group. It's the straightforward calculation we've just seen - the number of occurrences divided by the total opportunities.

Relative risk

Relative risk compares the likelihood of an event occurring in one group compared to another group. This helps us understand how much more (or less) likely something is to happen in different circumstances.

$\text{Relative risk} = \frac{\text{Risk in group A}}{\text{Risk in group B}}$

Worked example: medical treatment comparison

lightbulbExample

Worked Example: Comparing Cancer Treatment Effectiveness

A study found that:

New treatment: 15% probability of being ineffective
Old treatment: 35% probability of being ineffective

To find the relative risk of the old treatment compared to the new treatment:

Step 1: Convert percentages to decimals

New treatment risk = 0.15
Old treatment risk = 0.35

Step 2: Calculate relative risk Relative risk = $\frac{0.35}{0.15} = 2.33...$

Step 3: Interpret the result This means the old treatment is approximately 2⅓ times more likely to be ineffective compared to the new treatment.

Comparing risks between different groups

When comparing risks across different populations or conditions, we can gain valuable insights for decision-making. Risk comparison helps us understand the effectiveness of different approaches or identify factors that influence outcomes.

lightbulbExample

Worked Example: Exercise and Injury Risk Assessment

This table shows the relationship between warming up before exercise and muscle injuries. We can calculate:

For people who warmed up: Risk of pulling a muscle = $\frac{12}{12 + 84} = \frac{12}{96} = 0.125$

For people who didn't warm up: Risk of pulling a muscle = $\frac{45}{45 + 72} = \frac{45}{117} = 0.385$

Relative risk calculation: Relative risk = $\frac{0.385}{0.125} = 3.08$

Interpretation: This shows that people who don't warm up are about 3 times more likely to pull a muscle compared to those who do warm up.

chatImportant

Understanding Relative Risk Values

When relative risk > 1: The first group has higher risk than the second group
When relative risk < 1: The first group has lower risk than the second group
When relative risk = 1: Both groups have equal risk

Real-world application: emergency services

Risk assessment is crucial in public services. Consider emergency response data where different areas have varying success rates for emergency response times.

infoNote

Practical Application

Area M11 achieves an 86% success rate for responding to life-threatening calls within 8 minutes, while area M28 achieves only 63%. The relative risk calculation would show that M28 has a higher risk of delayed response compared to M11, which could inform resource allocation decisions.

This demonstrates how risk assessment helps in making informed decisions about resource distribution and policy implementation in real-world scenarios.

bookmarkSummary

Key Examination Tips

Always convert to decimals when comparing risks - this makes comparison much clearer than working with fractions.
Check your relative risk interpretation - if the relative risk is greater than 1, the first group has higher risk; if it's less than 1, the first group has lower risk.
Show all working clearly - examiners want to see your method, even if your final answer isn't perfect.
Read questions carefully - make sure you understand which group is being compared to which.
Consider real-world context - think about whether your calculated risk makes sense in the situation described.

bookmarkSummary

Key Points to Remember

Risk in probability is calculated using real data: number of occurrences divided by total trials
Absolute risk tells you the probability for one group; relative risk compares two groups
Converting risks to decimals makes comparison much easier
Relative risk greater than 1 means the first group has higher risk than the second group
Always show your working step-by-step in examinations
Risk assessment helps in making informed decisions in real-world situations

Using probability to assess risk (AQA GCSE Statistics): Revision Notes