Combined events and tree diagrams explained
Probability • Probability
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Key concepts
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Combined events and sample spaces
A combined event refers to the occurrence of two or more events together, for example 'A and B' or 'A or B'. The sample space lists all possible outcomes so that probabilities can be assigned and combined logically. Complete enumeration of the sample space prevents omission of outcomes and ensures probabilities sum to 1. Limiting factors include unequal likelihoods and hidden dependencies. Equal likelihood assumptions apply only when each outcome in the sample space has the same chance. Failure to state assumptions about replacement or independence leads to incorrect combined probabilities.
Independent events
Two events are independent when the occurrence of one event does not change the probability of the other. Mathematical expression: P(A and B) = P(A) × P(B) when A and B are independent. Independence justifies multiplying branch probabilities directly on a tree diagram. Limiting factors include mistaken independence when sampling without replacement or when events share underlying causes. Independence must be justified from the context or the process; never assume independence without evidence.
Dependent events and conditional probability
Dependent events occur when the probability of a later event changes because of an earlier outcome. Conditional probability notation P(B | A) indicates the probability of B given that A has occurred. The combined probability follows P(A and B) = P(A) × P(B | A). Limiting factors include failure to update probabilities after the first event and misreading the conditioning direction. Conditional probabilities require careful attention to what has already happened and how it affects remaining outcomes.
Tree diagrams as a representation
Tree diagrams represent sequential experiments by drawing branches for each possible outcome at each stage. Branch labels show the probability of that branch given the previous choices, and probabilities along a path multiply to give the probability of the sequence. Final outcomes appear as leaves whose probabilities are the product of branch probabilities along the path. Limiting factors include clutter for many stages and mislabelling branch probabilities. Tree diagrams require correct conditional probabilities on later branches when events are dependent, and accurate normalization so that branches from any node sum to 1.
Replacement versus no replacement
Replacement means returning an item to the population before the next draw so that probabilities remain the same for each draw. No replacement means removal of an item changes the composition of the population and therefore later probabilities. Replacement yields independence in repeated draws from the same set; no replacement yields dependence. Limiting factors include ignoring population size when assuming replacement or failing to adjust probabilities when sampling without replacement. Small populations change probabilities more markedly after each draw.
Multiplication rule and other representations
The multiplication rule states P(A and B) = P(A) × P(B | A). For independent events the conditional probability reduces to P(B), so P(A and B) = P(A) × P(B). Other representations include two-way tables, Venn diagrams and sample-space diagrams; each representation supports the same arithmetic but provides different visual clarity depending on the problem structure. Limiting factors include misuse of addition rules for 'and' events and confusing 'and' with 'or'. Proper selection of representation aids reasoning: trees suit sequential experiments, tables suit categorical combinations, and Venn diagrams suit overlapping events.
Key notes
Important points to keep in mind