CIE IGCSE Mathematics (0580) Relative and expected frequencies Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Relative and expected frequencies Study Notes
LEARNING OBJECTIVE
- Relative Frequency as an Estimate of Probability
Key Concepts:
- Relative Frequency as an Estimate of Probability
- Expected Frequencies and Estimates from Probability
Relative Frequency as an Estimate of Probability
Relative Frequency as an Estimate of Probability
Relative frequency is used when estimating the probability of an event based on experimental or observed results. It is calculated using:
$ \text{Relative Frequency} = \dfrac{\text{Number of times the event happens}}{\text{Total number of trials}}$
This is particularly useful when the actual probability is unknown or difficult to calculate theoretically.
Example:
A spinner was spun 50 times. It landed on red 18 times. Estimate the probability of landing on red.
▶️ Answer/Explanation
\( \text{Relative frequency} = \dfrac{18}{50} = 0.36 \)
So the estimated probability of landing on red is 0.36.
Example:
A biased coin is flipped 200 times. It lands on heads 130 times. Estimate the probability of getting tails.
▶️ Answer/Explanation
\( P(\text{heads}) = \dfrac{130}{200} = 0.65 \)
\( P(\text{tails}) = 1 – 0.65 = 0.35 \)
Expected Frequencies and Estimates from Probability
Expected Frequencies and Estimates from Probability
If an experiment is repeated a number of times, we can use probability to estimate how many times a particular event will occur. This is called the expected frequency.
$ \text{Expected Frequency} = \text{Probability} \times \text{Number of Trials} $
- Fair: All outcomes are equally likely.
- Biased: Some outcomes are more likely than others.
- Random: Each outcome is uncertain, but governed by probability.
Example :
A biased coin lands on heads with probability 0.6. The coin is tossed 150 times. Estimate how many times it will land on heads.
▶️ Answer/Explanation
Use expected frequency formula:
\( \text{Expected heads} = 0.6 \times 150 = \boxed{90} \)
Example :
A fair 6-sided die is rolled 240 times. How many times would you expect to get a 2?
▶️ Answer/Explanation
Since the die is fair, \( P(\text{2}) = \dfrac{1}{6} \)
\( \text{Expected frequency} = \dfrac{1}{6} \times 240 = \boxed{40} \)
Example :
In a factory, 5% of lightbulbs are faulty. A batch of 2000 bulbs is tested. Estimate the number of faulty bulbs expected in the batch.
▶️ Answer/Explanation
\( P(\text{faulty}) = 0.05 \)
\( \text{Expected faulty bulbs} = 0.05 \times 2000 = \boxed{100} \)