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Probability - Theoretical and Experimental Probability

Grade 9IGCSE

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

🔑Concepts

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Theoretical Probability: The likelihood of an event occurring based on mathematical reasoning, assuming all outcomes are equally likely.

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

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Sample Space: The set of all possible outcomes of an experiment, often denoted by 'S'.

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Probability Scale: All probabilities lie between 0 (impossible) and 1 (certain).

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Complementary Events: The probability of an event NOT occurring is 1 minus the probability that it does occur.

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

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

📐Formulae

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

Relative Frequency=Frequency of eventTotal number of trials\text{Relative Frequency} = \frac{\text{Frequency of event}}{\text{Total number of trials}}

P(A′)=1−P(A)P(A') = 1 - P(A) (Complementary Law)

Expected Frequency=P(Event)×Number of trials\text{Expected Frequency} = P(\text{Event}) \times \text{Number of trials}

∑P(x)=1\sum P(x) = 1 (The sum of probabilities of all possible outcomes is always 1)

💡Examples

Problem 1:

A fair six-sided die is rolled. What is the theoretical probability of rolling a prime number?

Solution:

P(Prime)=36=0.5P(\text{Prime}) = \frac{3}{6} = 0.5 or 12\frac{1}{2}

Explanation:

The sample space is {1, 2, 3, 4, 5, 6}, so the total outcomes are 6. The prime numbers in this set are {2, 3, 5}, which are 3 favorable outcomes. Divide favorable by total.

Problem 2:

A drawing pin is dropped 200 times. It lands 'point up' 140 times. Calculate the relative frequency of the pin landing 'point up'.

Solution:

Relative Frequency=140200=0.7\text{Relative Frequency} = \frac{140}{200} = 0.7

Explanation:

Experimental probability is found by dividing the number of times the event occurred (140) by the total number of trials (200).

Problem 3:

The probability that a bus is late is 0.15. If the bus runs 20 times a month, how many times would you expect it to be late?

Solution:

Expected Frequency=0.15×20=3\text{Expected Frequency} = 0.15 \times 20 = 3

Explanation:

To find the expected frequency, multiply the theoretical probability of the event (0.15) by the total number of trials or occurrences (20).

Problem 4:

A bag contains red, blue, and green marbles. The probability of picking a red marble is 1/3 and the probability of picking a blue marble is 1/4. Find the probability of picking a green marble.

Solution:

P(Green)=1−(13+14)=1−712=512P(\text{Green}) = 1 - (\frac{1}{3} + \frac{1}{4}) = 1 - \frac{7}{12} = \frac{5}{12}

Explanation:

Since the sum of all probabilities in a sample space must equal 1, subtract the sum of the known probabilities from 1 to find the remaining category.