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Probability - Theoretical and Experimental Probability

Grade 10IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Definition of Probability: A numerical measure of the likelihood that an event will occur, ranging from 0 (impossible) to 1 (certain).

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Sample Space: The set of all possible outcomes of an experiment.

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Theoretical Probability: Probability calculated based on reasoning and equally likely outcomes without conducting an experiment.

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Experimental Probability (Relative Frequency): Probability calculated based on the results of an actual experiment or trials.

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Law of Large Numbers: As the number of trials in an experiment increases, the experimental probability tends to get closer to the theoretical probability.

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Complementary Events: The probability that an event will not occur, denoted as P(A′)P(A'), where P(A)+P(A′)=1P(A) + P(A') = 1.

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Expected Frequency: The number of times an event is predicted to occur over a specific number of trials.

📐Formulae

Theoretical Probability: P(E)=Number of favorable outcomesTotal number of possible outcomesP(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}

Experimental Probability (Relative Frequency): Relative Frequency=Frequency of eventTotal number of trials\text{Relative Frequency} = \frac{\text{Frequency of event}}{\text{Total number of trials}}

Complementary Event: P(not A)=1−P(A)P(\text{not } A) = 1 - P(A)

Expected Frequency: Expected Frequency=P(E)×n\text{Expected Frequency} = P(E) \times n, where nn is the number of trials.

💡Examples

Problem 1:

A fair six-sided die is rolled. (a) What is the theoretical probability of rolling a prime number? (b) If the die is rolled 300 times, how many times would you expect to roll a prime number?

Solution:

(a) P(Prime)=36=0.5P(\text{Prime}) = \frac{3}{6} = 0.5 (b) Expected Frequency=0.5×300=150\text{Expected Frequency} = 0.5 \times 300 = 150

Explanation:

The prime numbers on a die are 2, 3, and 5 (3 outcomes). The total outcomes are 6. Theoretical probability is 3/63/6. To find the expected frequency, multiply this probability by the total trials (300).

Problem 2:

A spinner is spun 50 times and lands on 'Red' 12 times. Calculate the experimental probability of the spinner landing on Red.

Solution:

Experimental Probability=1250=0.24\text{Experimental Probability} = \frac{12}{50} = 0.24

Explanation:

Experimental probability is the ratio of the frequency of the specific outcome to the total number of trials performed.

Problem 3:

The probability that a seed germinates is 0.85. If 200 seeds are planted, how many are expected NOT to germinate?

Solution:

P(not germinate)=1−0.85=0.15P(\text{not germinate}) = 1 - 0.85 = 0.15; Expected=0.15×200=30\text{Expected} = 0.15 \times 200 = 30

Explanation:

First, find the probability of the complement (not germinating) by subtracting the given probability from 1. Then, multiply this result by the total number of seeds.