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In GCSE maths, this means knowing how to list combinations and sets of events, as well as using theoretical probability to find the likelihood of specified events.  \nSystematic Listing  \nPossible combinations can be found by listing. If you have the numbers 1, 2 and 3 and want to find out how many 3-digit numbers can be made, start by listing all the three-digit numbers beginning with 1, then 2, then 3: 123, 132, 213, 231, 312, 321. By systematically working through the options, we can be sure that no combinations have been missed.  \nProduct Rule (Higher Only)  \nWhen finding the number of possible combinations or combined outcomes, the product rule can be applied. The product rule is:  \nIf there are 􀝔 outcomes of Event 1 and 􀝕 outcomes of Event 2, then there 􀝔􀝕 combined  \noutcomes.  \nIn a mathematical problem, this means multiplying all the possibilities of multiple events together.  \nFor example, if you have 10 shirts and 8 pairs of trousers, then you could wear 􀫡􀫙 (10 × 8) different outfits. This applies for any number of events, as long as none of them are limited – if you have already chosen a shirt, then your possible choices of shirt reduce to 1. There are now only 􀫡 (1 × 8) possible outfits you could wear.  \nSimilarly, if you have 10 shirts, 8 pairs of trousers and 4 ties, then you could wear 􀫛􀫝􀫙(10 × 8 × 4) different outfits.  \nSystematic listing can be represented in different ways, including lists, tables and diagrams.  \nProbability Notation  \nThese are some key terms and ideas in probability – many of these are covered in more detail in the other Probability revision notes.  \n• Theoretical probability is the mathematical likelihood of something happening. It is always given a value between 0 and 1.  \n• An event is the outcome of an action or experiment.  \n• The sample space is all possible outcomes.  \n• The target outcome is the outcome that we want to find the probability of occurring.  \n• The possible outcomes are all the outcomes in the sample space.  \n• A population is the overall group of people, values or items that can be grouped by a common characteristic.  \n• A sample is a smaller sub-group of a population.  \nTypes of Events  \nEvents are independent if the probability of one outcome occurring does not affect the probability of another outcome occurring.  \nEvents are dependent if the probability of one outcome occurring does affect the probability of another outcome occurring.  \nEvents are mutually exclusive if they cannot both occur.  \nProbability Notation  \nAs well as key terms, we can use notation to describe events and their theoretical probability.  \n• 􀜲 (􀜣) means the probability of outcome A.  \n• 􀜲 (􀜣 ∪ 􀜤) means the probability of outcome A or outcome B.  \n• 􀜲 (􀜣 ∩ 􀜤) means the probability of outcome A and outcome B.  \n• 􀜲 (􀜤′) means the probability of not outcome B.  \nSet Notation  \nSet notation is used in mathematics to show a list of outcomes.  \nSet notation uses curly brackets {} known as braces. Everything listed inside the braces is an element of the set.  \n• ℤ is the set of integer numbers.  \n• ℕ is the set of natural number. This is the set of positive integers {1, 2, 3, 4,…} .  \n• 􀟦 is the universal set. The universal set is the whole set we are considering in a question or problem. It is used frequently in Venn diagrams.  \n• A ⊂ B means that A is a subset of B. This means every element in set A must also be an element in set B.  \nSet notation can be used to describe a situation. For example,  \n􀜼 = {􀝔 ∶ 􀝔 􀝅􀝏 􀜽 􀝂􀜽􀜿􀝐􀝋􀝎 􀝋􀝂 20}  \nmeans 􀜼 is the set of numbers 􀝔 such that 􀝔 is a factor of 20 , i.e. 􀜼 is the set of factors of 20. It could also be written as 􀜼 = {1, 2, 4, 5, 10, 20} .  \nTables and Grids  \nTypes of tables and grids can be used to show the outcomes of combined events. This is when two events have o","cbCaigPIqoyv0iUZ","https://ap.wps.com/l/cbCaigPIqoyv0iUZ","pdf",563855,11,"English","# Enumeration\n## Systematic listing\n## Product rule (higher only)\n# Probability notation and key terms\n# Types of events\n# Set notation\n# Tables, grids, and diagrams\n## Tree diagrams\n## Venn diagrams\n## Intersection and union","[{\"question\":\"How does systematic listing help with probability questions?\",\"answer\":\"Systematic listing means working through all possible combinations in an ordered way so you can be confident that none are missed.\"},{\"question\":\"What does the product rule say for combined outcomes?\",\"answer\":\"If Event 1 has x outcomes and Event 2 has y outcomes, then the combined outcomes are x × y, provided none of the events are restricted.\"},{\"question\":\"What is the difference between independent and dependent events?\",\"answer\":\"Independent events are unaffected by each other, while dependent events change the probability because one event’s outcome affects the other.\"}]","GCSE Maths - Probability - Enumeration, Venn Diagrams, Tree Diagrams and Tables | PDF",1792977549,28]