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Statistics and Probability - The outcomes of random events (Probability)

Grade 7IGCSE

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

🔑Concepts

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Probability Scale: Probability is measured on a scale from 0 (impossible) to 1 (certain). It can be expressed as a fraction, decimal, or percentage.

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Outcomes and Sample Space: An outcome is a possible result of an experiment. The sample space is the set of all possible outcomes.

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

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Complementary Events: The sum of the probability of an event happening and it not happening is always 1. These are called mutually exclusive outcomes.

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Relative Frequency (Experimental Probability): The ratio of the number of times an event occurs to the total number of trials performed in an experiment.

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

📐Formulae

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

P(Not A)=1−P(A)P(\text{Not A}) = 1 - P(A)

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

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

💡Examples

Problem 1:

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

Solution:

P(Prime)=36=12P(\text{Prime}) = \frac{3}{6} = \frac{1}{2} or 0.50.5

Explanation:

The sample space is {1, 2, 3, 4, 5, 6}, so there are 6 possible outcomes. The prime numbers on a die are 2, 3, and 5 (3 successful outcomes). Dividing successful outcomes by total outcomes gives 3/6.

Problem 2:

The probability that it will rain tomorrow is 0.27. What is the probability that it will not rain?

Solution:

P(No rain)=1−0.27=0.73P(\text{No rain}) = 1 - 0.27 = 0.73

Explanation:

Since raining and not raining are complementary events, their probabilities must add up to 1. Subtracting the probability of rain from 1 gives the probability of no rain.

Problem 3:

A bag contains 5 red, 3 blue, and 2 green marbles. If a marble is picked at random, what is the probability it is NOT blue?

Solution:

P(Not Blue)=710P(\text{Not Blue}) = \frac{7}{10}

Explanation:

First, find the total number of marbles: 5+3+2=105 + 3 + 2 = 10. The number of marbles that are not blue is 5(red)+2(green)=75 (red) + 2 (green) = 7. Thus, the probability is 7/10.

Problem 4:

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

Solution:

Relative Frequency=1250=0.24\text{Relative Frequency} = \frac{12}{50} = 0.24

Explanation:

Relative frequency is calculated by dividing the observed frequency (12) by the total number of trials (50).