The Basics of Probability (Leaving Cert Mathematics): Revision Notes
Expected Frequency
Explanation
Expected frequency is a concept in probability and statistics that predicts how often an event should occur based on theoretical probabilities. It is calculated using the formula:
Steps to Calculate Expected Frequency
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Determine the Probability of the Event: Identify the likelihood of the event happening. This could be given as a fraction, decimal, or percentage.
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Identify the Total Number of Trials: Determine how many times the experiment or observation will be conducted.
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Apply the Formula: Multiply the probability of the event by the total number of trials to find the expected frequency.
Expected frequency is commonly used in hypothesis testing, such as in the chi-squared test, and helps compare actual experimental results with theoretical predictions.
Worked Examples
Example 1: Tossing a Fair Coin
Problem: A coin is tossed 100 times.
How many times would you expect it to land on heads?
Solution:
- Step 1: Probability of heads = 0.5 (since the coin is fair).
- Step 2: Total number of trials = 100.
- Step 3: Expected frequency = 0.5 × 100 = 50
Answer: The expected frequency of getting heads is 50.
Example 2: Rolling a Die
Problem: A six-sided die is rolled 120 times.
How many times would you expect to roll a 4?
Solution:
- Step 1: Probability of rolling a 4 = 1/6
- Step 2: Total number of trials = 120.
- Step 3: Expected frequency = 1/6 × 120 = 20
Answer: The expected frequency of rolling a 4 is 20.
Example 3: Real-World Application (Traffic Lights)
Problem: In a survey, it's found that 30% of vehicles at an intersection run a red light. If 500 vehicles pass through the intersection, how many would you expect to run a red light?
Solution:
- Step 1: Probability of running a red light = 0.3.
- Step 2: Total number of trials = 500.
- Step 3: Expected frequency = 0.3 × 500 = 150
Answer: The expected frequency of vehicles running a red light is 150.
Summary
- Expected Frequency Formula:
- Helps compare observed data with theoretical probabilities.
- Applies to various contexts, such as dice rolls, coin tosses, and real-world surveys.