WEBVTT 00:00:00.500 --> 00:00:03.868 Welcome to GCSE Edexcel Maths revision. 00:00:04.018 --> 00:00:08.435 Topic M 37: Sample spaces, sets and Venn diagrams. 00:00:08.585 --> 00:00:11.379 This video covers Foundation and Higher tiers. 00:00:14.033 --> 00:00:16.922 A sample space lists all possible outcomes. 00:00:17.072 --> 00:00:22.847 For one coin toss it is {H, T}; for two tosses it is {H H, H T, T H, T T}. 00:00:22.997 --> 00:00:26.111 H T and T H are distinct ordered outcomes. 00:00:28.767 --> 00:00:31.934 A table is useful when combining two experiments. 00:00:32.084 --> 00:00:37.865 Put one experiment's outcomes along each side, and fill every cell with the resulting pair or value. 00:00:38.015 --> 00:00:40.745 Sample-space table for sums of two dice 00:00:45.400 --> 00:00:49.997 Worked example: Two fair dice have 36 equally likely ordered pairs. 00:00:50.147 --> 00:00:58.754 Sum 7 occurs in 6 cells, so probability of open bracket sum 7 close bracket equals 6 over 36 equals 1 over 6 . 00:01:01.433 --> 00:01:04.632 Possible sums 2 to 12 are not equally likely. 00:01:04.782 --> 00:01:10.551 There is one way to make 2 and six ways to make 7, so counting the eleven sums equally would be wrong. 00:01:13.233 --> 00:01:18.821 For a coin and a three-outcome spinner, a table has 2 multiplied by 3 equals 6 cells. 00:01:18.971 --> 00:01:25.242 Equal probabilities in the cells require each device to have equally likely outcomes and independence. 00:01:27.900 --> 00:01:30.033 A set is a collection of elements. 00:01:30.183 --> 00:01:31.805 The universal set, 00:01:31.955 --> 00:01:34.045 often written the universal set , 00:01:34.195 --> 00:01:38.697 contains every element under consideration. not A is the complement of A, 00:01:38.847 --> 00:01:42.498 containing elements in the universal set but outside A. 00:01:45.167 --> 00:01:51.128 A intersection B means elements in both sets; A union B means elements in A or B or both. 00:01:51.278 --> 00:01:53.751 Or in a union includes the overlap. 00:01:53.901 --> 00:01:56.291 French and Spanish Venn diagram 00:02:00.967 --> 00:02:04.389 Put elements belonging to both sets in the overlap first. 00:02:04.539 --> 00:02:10.052 Then fill A-only and B-only regions, and finally the region outside both circles. 00:02:12.733 --> 00:02:14.056 For two sets, 00:02:14.206 --> 00:02:30.141 number of elements in open bracket A union B close bracket equals number of elements in open bracket A close bracket plus number of elements in open bracket B close bracket minus number of elements in open bracket A intersection B close bracket . 00:02:30.291 --> 00:02:34.601 The overlap is subtracted once because it was included in both totals. 00:02:37.267 --> 00:02:43.287 Worked example: Of 30 students, 18 study French, 12 study Spanish and 5 study both. 00:02:43.437 --> 00:02:50.485 French only equals 13, Spanish only equals 7, either language equals 25 and neither equals 5. 00:02:53.167 --> 00:02:54.575 For the same group, 00:02:54.725 --> 00:03:08.867 probability of open bracket French and Spanish close bracket equals 5 over 30 equals 1 over 6 and probability of open bracket French or Spanish close bracket equals 25 over 30 equals 5 over 6 . 00:03:09.017 --> 00:03:11.555 Both use the full group as denominator. 00:03:14.233 --> 00:03:22.029 To find probability of open bracket not A intersection B close bracket , use the B-only region, not everything outside A. 00:03:22.179 --> 00:03:25.560 Reading the symbols carefully avoids choosing the wrong region. 00:03:28.233 --> 00:03:35.645 If a question uses three sets, work from the central triple overlap outwards and use the total to check the remaining region. 00:03:35.795 --> 00:03:38.313 Follow all given counts consistently. 00:03:40.967 --> 00:03:44.293 Counts must be non-negative and sum to the stated total. 00:03:44.443 --> 00:03:48.874 A negative only region is a sign that the data were interpreted incorrectly. 00:03:51.533 --> 00:03:55.072 That completes Sample spaces, sets and Venn diagrams. 00:03:55.222 --> 00:03:58.567 Revisit the notes and test yourself on the revision website.