Using probability to assess risk Revision Notes for Edexcel GCSE Statistics

Using probability to assess risk

What is risk assessment?

Risk assessment involves using probability calculations to understand how likely accidents, problems, or unwanted events are to occur. In mathematics, we can calculate the risk of any event by analysing data from real-world situations.

The risk of an event is essentially its experimental probability - it tells us how often something actually happens based on collected data rather than theoretical calculations.

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Experimental vs Theoretical Probability

Experimental probability uses real-world data from actual trials and observations, while theoretical probability is based on mathematical models and expected outcomes. Risk assessment relies on experimental probability because it reflects what actually happens in practice.

Basic risk formula

The fundamental formula for calculating risk is:

$\text{Risk of event} = \frac{\text{Number of trials in which event happens}}{\text{Total number of trials}}$

This gives us a decimal value between 0 and 1, where:

0 means the event never happens
1 means the event always happens
Values closer to 1 indicate higher risk

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Converting risk to percentages makes it easier to communicate with others. Simply multiply your decimal result by 100. For example, a risk of 0.12 becomes 12%.

Worked example: Basketball injuries

Let's examine injury data from basketball players to understand risk calculation:

Injury type	Fingers	Feet	Knees
Number of injuries	6	8	3
Games played	50	46	50

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Worked Example: Calculating Basketball Injury Risk

Step-by-step calculation for finger injuries:

Risk of finger injury = $\frac{\text{Number of finger injuries}}{\text{Total games played}}$

Risk of finger injury = $\frac{6}{50} = 0.12$

This means there's a 12% chance of a finger injury in any given game.

For the other injury types:

Feet: $\frac{8}{46} = 0.17$ (17% risk)
Knees: $\frac{3}{50} = 0.06$ (6% risk)

Converting to decimals makes it easier to compare different risks - we can immediately see that foot injuries are the most common, followed by finger injuries, then knee injuries.

Understanding absolute and relative risk

Absolute risk

This is simply the probability of an event happening - the straightforward calculation we've been doing. It gives us the actual likelihood of something occurring.

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Absolute risk provides the baseline measurement for any risk assessment. It answers the question: "What are the chances this will happen to me?"

Relative risk

This compares the risk between different groups. It tells us how many times more likely an event is to happen in one group compared to another.

Relative risk formula: $\text{Relative risk} = \frac{\text{Risk for those in the group}}{\text{Risk for those not in the group}}$

Worked example: Medical treatments

Consider a study comparing two cancer treatments:

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

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Worked Example: Comparing Treatment Effectiveness

Question: What is the relative risk of the old treatment being ineffective compared to the new treatment?

Step-by-step solution:

First, identify the risks:
- Risk for old treatment = $0.35$
- Risk for new treatment = $0.15$
Apply the relative risk formula: Relative risk = $\frac{0.35}{0.15}$
Calculate: $\frac{0.35}{0.15} = \frac{35}{15} = \frac{7}{3} = 2.33...$
Interpret the result: The relative risk is $2\frac{1}{3}$ , meaning the old treatment is 2⅓ times more likely to be ineffective than the new treatment.

Key problem-solving strategies

When calculating risk:

Always identify what event you're measuring
Count the number of times the event occurred
Find the total number of opportunities for the event
Use the basic risk formula
Convert to decimals for easy comparison

When comparing risks:

Calculate the absolute risk for each group first
Decide which group you're comparing to which
Use the relative risk formula
Interpret your answer in context

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Common exam traps to avoid:

Don't confuse the numerator and denominator in risk calculations
Remember that relative risk compares groups, not individual probabilities
Always check your decimal calculations carefully
When interpreting relative risk, specify which group has the higher risk

Real-world applications

Risk assessment is used extensively in:

Healthcare: Comparing treatment effectiveness and side effects
Insurance: Determining premiums based on accident likelihood
Safety planning: Identifying which activities need better protection
Emergency services: Allocating resources based on response time data

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Understanding these calculations helps you make informed decisions about everyday risks and evaluate claims made in news reports or advertisements. When you see statistics in the media, you can now assess whether the risks are being presented fairly and accurately.

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Key Points to Remember:

Risk is calculated as the number of events divided by total trials
Convert risks to decimals to make comparisons easier
Relative risk tells you how many times more likely an event is in one group versus another
Always interpret your numerical answers in the context of the real-world situation
Risk assessment helps us make better decisions based on actual data rather than guesswork

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