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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:
|[pic] |What percent of students can not take bio-research? |
| |Are the events ‘taken Stats’ and ‘taken Comp’ disjoint or mutually |
| |exclusive? |
| |What percent of the students have taken only one of the prerequisites |
| |for Bio-research? |
| |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: |[pic] |
|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 |Total |
| | | |soccer | |
|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 )=
c) What is the probability that if we select a person at random from that group he has been an elite soccer player and has arthritis? P( elite soccer player and arthritis)=
d) What is the probability that if we select a person at random from the elite soccer player group, he has arthritis? This would be P(arthritis/ elite soccer player) =
it also can be written as P(having arthritis given that he has been an elite soccer player)

3. The notion of independence
Two events A and B are said to be independent if
P(A and B) = P(A)P(B)
That can also be expressed as P(A)= P(A/B) meaning that the fact that B happens does not change the probability of A happening.
In the example of A, compare P(elite soccer player and arthritis) with
P(elite soccer player) * P(arthritis)
Based on that comparison, do you think that having arthritis in the hip or knee is independent of being an elite soccer player? ______
If we assume that events are independent we calculate the probability that both happen as the product of their probabilities
4. Conditional probabilities in the context of medical diagnosis: False positives and false negatives.
Diagnosis through blood tests is not always correct, there is always a certain possibility of ‘false positives’ (people who are healthy but are diagnosed as sick) and ‘false negatives’ (people whose test is OK but they are actually sick).
The following table corresponds to the results of the Eliza test for Aids for one million people

| | True health status |
|Test result |B1: Carry AIDS Virus |B2: Do not carry AIDS virus |Totals |
|A1: Test result is Positive |4885 |73630 |78515 |
|A2: Test result is Negative |115 |921370 |921485 |
|Total |5000 |995000 |100000 |

a) What is the probability that a person that got the ELIZA test positive really carries the AIDS virus?
P(Carries AIDS virus/ ELIZA test is +) =
b) What is the probability that a person who really carries the AIDS virus tests positive in the ELIZA test? P( + in ELIZA / carries AIDS virus) =
c)What is the probability of a ‘false positive’ or P(+ in ELIZA/ does not carry AIDS virus) =
d)What is the probability of a ‘false negative’ or P(- in ELIZA /carries AIDS virus)=

5. The Birthday problem. Using the ‘probability of the complement’ and the notion of independence
|This is a classic problem in probability. You can find tons of references in the |[pic] |
|internet (the graph below is |So the probability of having at least two people with the same |
|fromhttp://www.mste.uiuc.edu/reese/birthday/explanation.html) . In a group of n |birthday will be |
|people (class, party etc.) what is the probability that at least two people share the|1-[pic]* [pic]*[pic]*[pic]*………….*[pic] |
|same birthday (month and day)? For example, in a class of 40 students what is the | |
|probability that we have some repeated birthday? Actually the probability is greater| |
|than 0.5 if there are 23 people of more in the group. | |
|The way we calculate this is by working with the complement, i.e. what is the | |
|probability that everybody has a different birthday? Probability that all birthdays | |
|are different: The first person can have her/his birthday on any day, but the second | |
|person to be different needs to have his/her birthday on one of the 364 remaining | |
|days, and we reason in the same way for the remaining people | |
|[pic]* [pic]*[pic]*[pic]*…………*[pic] | |