Topic:Review for Business Statistics 1 - Probabilities
From SharedExperienceProject
Contents |
Objective
This page is written to help students prepare for the midterm and final exams for Business Statistics 1.
It does not replace the need to review each topic or do the practice problems in each topic. Rather, it supplements those learning activities with a summary of steps you should follow as you study, and with some additional practice problems.
Preparing for Probability Problems
This section is intended to help you prepare for exam questions on probabilities.
Basic Preparation Steps
1. Review the following topics, being sure you can work the examples and practice problems in each:
- Introduction to Probability
- More on Probability
- Remember my emphasis on contingency tables (e.g. Example 8) vs. Venn Diagrams (e.g. Example 7)
2. Do the following:
- Test Your Knowledge: Getting Probabilities from Contingency Tables (below on this page)
- Test Your Knowledge: Building Contingency Tables from the Given Information (also below)
- Test Your Knowledge: Tying it All Together (also below)
3. Review Quiz 3
Test Your Knowledge: Getting Probabilities from Contingency Tables
Note: Complete solutions will only be provided for selected problems. Answers will be provided for others. We can cover some of the remaining during the review sessions.
Practice Problem 1
What are the following marginal probabilities for the contingency table given below?
a) Probability of event A occuring
b) Marginal probability P(B)
c) Probability that A won't occur
d) P(not B)
| Solution |
|---|
|
The marginal probabilities are those corresponding to the totals in a contingency table a) P(A) = 120/200 = 0.600 b) P(B) = 60/200 = 0.300 c) P(not A) = 80/200 = 0.400 d) P(not B) = 140/200 = 0.700 Note: P(not A) and P(not B) could also be obtained as the complement of P(A) and P(B), respectively |
Practice Problem 2
For the contingency table below, how many values fall into the following categories?
a) A and B
b) A and not B
c) not A and B
d) not A and not B
| Solution |
|---|
|
a) 36 b) 84 c) 24 d) 56 |
Practice Problem 3
For the above contingency table, what are the following probabilities?
a) P(A and B)
b) P(A and not B)
c) P(not A and B)
d) P(not A and not B)
| Solution |
|---|
|
These are just the number in that class divided by the total number: a) P(A and B) = 36/200 = 0.180 b) P(A and not B) = 84/200 = 0.420 c) P(not A and B) = 24/200 = 0.120 d) P(not A and not B) = 56/200 = 0.280 |
Practice Problem 4
What are the following conditional probabilities for the contingency table given below?
a) Probability of A given B?
b) Probability of A given not B?
c) P(not A given B)?
d) P(not A given not B)?
| Solution |
|---|
|
a) P(A | B) = 36/60 = 0.600 b) P(A | not B) = 84/140 = 0.600 c) P(not A | B) = 24/60 = 0.400 d) P(not A | not B) = 56/140 = 0.400 Note: c) is the complement of a), and d) is the complement of b) - could have solved that way too |
Test Your Knowledge: Building Contingency Tables from the Given Information
Practice Problem 5
Imagine that you run the sports centre at Mount Royal College. You carry out a survey of several classes at the college to find out that 20% of the students work out before classes, 50% work out after classes, and 10% (the hard core students) work out both before and after class.
Set up the contingency table for this situation.
| Solution |
|---|
|
First, define the relevant classes of students: B = works out before A = works out after Then, write down the probabilities you are given - you should always do this: P(B) = 0.20 P(A) = 0.50 P(B and A) = 0.10 Then, set up the contingency table and place the given data in the table. To do this, you can arbitrarily assume some size of the sample. Let's use a nice round number like 100: Then you can fill in the rest of the table: This is the simplest of these kinds of problems and you should be able to do it easily. You might also recognize it as Example 8 from the topic More on Probability that we did in class way back when we studied probabilities. With the table set up, you should be able to solve many of probability problems of the type outline above under Getting Probabilities from Contingency Tables. |
Practice Problem 6
Refer to the "Probability Problems 1" posted on Blackboard from when we first covered this material.
Set up the contingency tables for each of the following of those problems:
- Challenge Problems - Question 1
- Challenge Problems - Question 2
- Challenge Problems - Question 3
Use the following steps in each of the above cases:
- Write down the relevant events, giving them meaningful abbreviations
- Write down the given probabilities, in terms of the events
- Draw the contingency table, often assuming a sample size like 100 or 1000
- Use the given probabilities to fill in the rest of the table
- Solve for remaining probabilities
The solutions are also posted on Blackboard under "Probability problems 1 - solutions - updated.pdf"
Practice Problem 7
Refer again to the "Probability Problems 1" posted on Blackboard from when we first covered this material.
Set up the contingency table for the following problem:
- Probability Concepts Problems - Question ONE
The solution is posted on Blackboard under "Getting the table for Prob Concepts ONE.PDF"
Test Your Knowledge: Tying it All Together
Although you should do all of the assigned problems (in Shex and on Blackboard), you might want to start with the following.
Practice Problem 8
Practice problems about "High Test College Caffeine Analysis" were handed out in class periods. The solutions are on Blackboard under "Probability problems 2 - solutions.PDF".
Solve it again for practice.
Practice Problem 9
You might also try Practice Problem 1 in the topic More on Probability
Practice Problem 10
In class we also saw a handout entitled "SAMPLE PROBABILITY QUESTION". The solution is posted on Blackboard under "Probability Problems 3".
These are great practice.
Summary Thoughts
- In these topics we took a first look at the concept of probability
- In Introduction to Probability, we started with a review of experiments, events and probabilities, and we looked at different definitions of probabilities
- Then we got into contingency tables and how they can be used to represent and solve probability problems
- In More on Probability we used Venn diagrams to understand where these equations come from and we took a deeper look at how to solve probabililty problems
- We looked at marginal, joint and conditional probabilities and introduced the following equations for the latter two:
P(A | B) = P(A and B) / P(B) (1)
P(A and B) = P(A | B) x P(B) (2)
- We also looked at the additive rule which is written as follows:
P(A or B) = P(A) + P(B) - P(A and B) (3)
- We looked at the following test for independent events, where equation (2) above takes the form:
P(A and B) = P(A) x P(B)
- We also looked at the special case of mutually exclusive events, where equations (1) and (3) take the form:
P(A and B) = 0 and
P(A or B) = P(A) + P(B)
- Finally, we also looked at the concept of complementary probabilities:
