In simple terms
A friendly intro before the formal notes — no formulas yet.
The Rules of Chance
Probability measures the likelihood of an event, from impossible (0) to certain (1). Understanding the rules for combining events is key to solving problems in statistics.
Think of a weather forecast. The chance of rain is a probability. The chance of 'rain' and the chance of 'no rain' must add up to 100%. If we consider 'rain' and 'strong wind', they could happen together (not mutually exclusive) and one might make the other more likely (not independent).
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P(A) is between 0 and 1; P(not A) = 1 − P(A).
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Mutually exclusive: P(A ∪ B) = P(A) + P(B).
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Independent: P(A ∩ B) = P(A) × P(B).
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Tree diagrams organise conditional branches systematically.
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Key formulas
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Full topic notes
Formal explanation with the rigour you need for the exam.
Fundamental Concepts of Probability
Probability is a numerical measure of the likelihood that an event will occur. It is always a value between 0 (impossibility) and 1 (certainty). The set of all possible outcomes of an experiment is called the sample space. An 'event' is a specific outcome or a collection of outcomes from the sample space. For any event A, its complement, denoted or , represents the event that A does not occur.
The Complement Rule:
Combining Events: Addition Rules
Often, we are interested in the probability of one event or another occurring. This is the 'union' of events, denoted . Sometimes, events cannot happen together; these are called mutually exclusive events. For example, when rolling a single die, the events 'roll a 6' and 'roll a 3' are mutually exclusive.
Addition Rule for Mutually Exclusive Events:
If two events can happen at the same time (they are not mutually exclusive), we must use the general addition rule. This rule prevents 'double counting' the probability of their intersection, denoted , which means 'A and B'. For example, drawing a 'King' and drawing a 'Heart' from a deck of cards are not mutually exclusive, as you can draw the King of Hearts.
General Addition Rule:
Mutually exclusive events have no overlap: .
Use the simple addition rule only for mutually exclusive events.
Always use the general addition rule if events might overlap. If they are mutually exclusive, the term will be zero anyway.
Independent Events and Conditional Probability
Two events are independent if the outcome of one has no effect on the outcome of the other. For example, flipping a coin twice; the result of the first flip does not influence the second. When events A and B are independent, the probability of them both occurring is simply the product of their individual probabilities.
Multiplication Rule for Independent Events:
When events are not independent, they are called dependent. This leads to the idea of conditional probability: the probability of an event A occurring, given that event B has already occurred. This is written as . For example, the probability of drawing a second King from a deck of cards is dependent on whether the first card drawn was a King.
Conditional Probability Formula:
A common exam question asks you to determine if two events are independent. To do this, calculate , , and from the information given. Then, check if is equal to . If they are equal, the events are independent. If not, they are dependent. Do not just assume independence unless the question explicitly states it or the context (e.g., 'with replacement') makes it obvious.
Worked examples
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In a group of 30 students, 15 study Chemistry, 20 study Physics, and 6 study neither subject. A student is selected at random. Find the probability that the student studies both Chemistry and Physics. Let C be the event a student studies Chemistry and P be the event a student studies Physics.
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We are given: , , , and . The number of students studying at least one subject is the total minus those studying neither: . We can now use probabilities. , , . We use the general addition rule: . Rearranging to find the intersection: . Substituting the values: . . The probability that a student studies both subjects is .
A bag contains 5 red balls and 4 blue balls. A ball is drawn at random and not replaced. A second ball is then drawn. Find the probability that: (i) both balls are red, (ii) the second ball is blue.
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This is a 'without replacement' problem, so events are dependent. A tree diagram is a useful tool. Let be the event the first ball is red, and be the event the first is blue. Similarly for the second draw (). , . The second-stage probabilities are conditional:
$P(R_2 R_1) = 4/8$ (4 red left out of 8 total) $P(B_2 R_1) = 4/8$ (4 blue left out of 8 total) $P(R_2 B_1) = 5/8$ (5 red left out of 8 total) $P(B_2 B_1) = 3/8$ (3 blue left out of 8 total)
How it all connects
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Glossary
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Revision flashcards
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What is the range of a probability value?
A probability must be between 0 and 1, inclusive. .
Key takeaways
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Mutually exclusive events have no overlap: .
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Use the simple addition rule only for mutually exclusive events.
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Always use the general addition rule if events might overlap. If they are mutually exclusive, the term will be zero anyway.
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