Download 5 Solved Problems on Notion of Independence - Examination 4 | MATH 1530 and more Exams Statistics in PDF only on Docsity! Math1530 - Interesting things related to probability We have already seen: Random experiments, random phenomena, outcomes, sample space, events and probability. Probabilities are numbers between 0 and 1 that are assigned to events and have the following characteristics: a) The probability of the whole sample space is 1 b) P(A does not occur) = 1- P(occurs) c) If A and B are disjoint (mutually exclusive) , then P(A or B occur) = P(A) + P (B) Now we will see some interesting tools, ideas and applications of probability 1. Venn Diagrams. Useful tools to depict events and organize information. Example : In DeVeaux & Velleman ‘Intro Stat’ we find the following story: “ A university requires its biology majors to take a course called bio-research. The prerequisite for this course is that students must have taken either a Statistics course or a computer course. By the time they are juniors, 52% of the Biology majors have taken Statistics, 23% have had a computer course and 7% have done both” . The following Venn-diagram can be used to describe the situation and make the analysis easier: a) What percent of students can not take bio- research? b) Are the events ‘taken Stats’ and ‘taken Comp’ disjoint or mutually exclusive? c) What percent of the students have taken only one of the prerequisites for Bio- research? d) If we select a student at random what is the probability that he/she has taken Stats but not the computer course? You can form also Venn Diagrams with more than 2 events. Example: In the study of osteoporosis, bone mineral density is measured in the spine and both hips. When a person has bone mineral density below a certain level he/she is diagnosed as having osteoporosis on a certain location. Osteoporosis can be present in just one or all the 3 locations. Describe who would be located in each region of the graph. 2. Calculating probabilities and conditional probabilities from two way tables Example 1 : Two-way tables are tables that contain information about two variables. For example, on page 137 (Chapter 7 ) of Moore’s The Basic Practice of Statistics we find the following table about knee/hip arthritis on men in their mid 50’s Elite soccer Non elite soccer Did not play soccer Total Arthritis 10 9 24 43 No arthritis 61 206 548 815 Total 71 215 572 858 a) What is the probability that if we select a person at random from that group, he has been an elite soccer player? This would be P(elite soccer player)= b) What is the probability that if we select a person at random from that group, he has arthritis? This would be P(arthritis )=
