Relative frequency offers a practical approach to understanding probability, bridging theoretical maths with real-world observations. It's calculated by dividing the number of times an event occurs by the total number of trials, providing a tangible estimate for the probability of an event.
Understanding Relative Frequency
- Definition: The ratio of the number of times an event has occurred to the total number of trials or experiments.
- Importance: Serves as an empirical estimate of the probability of an event, grounding abstract probability concepts in observable outcomes.
Calculating Relative Frequency
Given a scenario, the relative frequency is obtained through:
Image courtesy of House of Math
Worked Examples
Example 1: Dice Roll
Problem: A six-sided die is rolled 120 times, landing on number 3, 20 times.
Solution:
1. Identify Occurrences: Event = rolling a 3, occurred 20 times.
2. Total Trials: 120 rolls.
The relative frequency suggests a chance of rolling a 3, aligning with the theoretical probability for a fair die.
Example 2: Coin Toss
Problem: A coin is tossed 200 times, yielding 97 heads.
Solution:
1. Identify Occurrences: Heads came up 97 times.
2. Total Trials: 200 tosses.
This indicates a roughly 48.5% chance of getting heads, close to the expected 50%.
Example 3: Drawing Cards
Problem: From a deck of 52 cards, a card is drawn 260 times with replacement, drawing a heart 70 times.
Solution:
1. Identify Occurrences: Heart drawn 70 times.
2. Total Trials: 260 draws.
The relative frequency of drawing a heart is approximately 26.9%, slightly lower than the theoretical 25% (or ).
Real-Life Applications
- Surveys and Studies: Relative frequency is used to analyse outcomes in surveys, such as the preference for a particular product.
- Quality Control: In manufacturing, the relative frequency of defects can help in assessing the quality of production.
Importance of Sample Size
A larger sample size yields a relative frequency closer to the theoretical probability, demonstrating the law of large numbers.
