Discrete outcomes can be counted – How many TVs in your house? By contrast, continuous outcomes are typically measured – How much do you weigh? Chapter 7 focuses on the probabilities of many discrete outcomes.
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Most of the chapter exercises in the text give you a complete probability distribution to work from, but what if you are only presented with frequencies and need to construct your own distribution? I’ll show you how in this video. I also check to see that the two requirements of a probability distribution have been satisfied.
There are two requirements that must be satisfied in order to say that we have a proper distribution of a discrete random variable.
To find the average (or the EXPECTED VALUE as it’s called in this chapter) of a list of outcomes when those outcomes are not equally likely to occur requires a new approach. This is a walk through of the expected value and its meaning.
The variance of a probability distribution also differs (in its calculation) from what you learned in Chapter 4. I extend the tabular approach that I used to find the expected value (previous video) so be able to find the variance of x. The standard deviation of x is simply the square root of the variance (just like in Chapter 4)
It’s not much of a shortcut, but some exams will ask you to use the shortcut formula specifically, so I demonstrate it here using the same data from question 1c)
Here I solve a full Chapter 7 exam question. First , I look at the language used in the question to identify which chapter's techniques are appropriate. Once this is determined to be a discrete probability distribution problem, I use a probability tree (learned in Chapt.6) to find all possible outcomes as well as their probabilities... to construct the Discrete Probability Distribution.
Now that we have the probability distribution, finding the probabilities of specific outcomes couldn't be any easier. If you can add, you can do this!
Time for the tabular approach to finding the expected value of x ( I use a table to keep track of my calculations). The order of operations comes into play here - Remember P.E.D.M.A.S. from high school math?
Once again, I extend the tabular approach that I used to find the expected value (previous video) so be able to find the variance of x. The standard deviation of x is simply the square root of the variance (just like in Chapter 4)
Your textbook can be confusing when it tries to explain the Laws of Expected Value. It can be easier to understand what's happening if you think about them as the Laws of THE NEW Expected Value. I explain in more detail in this video.
Test your understanding of Discrete Probability Distributions - each question is accompanied by a mini-video lecture showing you how I decided which solution was the correct one.
One more exam-style question before we get to the Binomial section of Chapter 7. This solution requires the use of the probability tree to find the probability distribution of x.
A Binomial Experiment is a special kind of discrete probability distribution with it’s own probability formula and tables. How do we know a question requires a binomial solution? It must satisfy the following conditions…
The binomial formula looks nasty, but it's easy to use once you learn how the different parts work. Your calculator will do most of the heavy work for you, and all you'll need are three numbers that will be given in the question: n, p, and x.
Knowing how to use the binomial tables is a MUST for the test! They can save you critical time compared with using the binomial formula. The tables however are not intuitive and so you'll first need to understand what kind of question the tables are designed to answer. Once that's clear, you can rework most binomial problems so that the table can solve them quickly.
When a discrete variable has only TWO possible outcomes (per observation), then it is a Binomial variable. The calculations of probabilities, expected value and variance are all different from the ones we just covered (for discrete probability distributions). I walk you through a question here that covers all of the new methods introduced with binomial variables.
This is a typical binomial distribution exam question. It's a trick question - the binomial 'setting' changes with each new question (a, b, c..) - so I show you what needs to be done to handle that. Binomial tables are introduced here and I show how they can be used as a much needed shortcut through the typically long calculations that binomial questions require.
More binomial, more probability questions, more work with the values on the binomial tables.
Test your understanding of Binomial Distributions - each question is accompanied by a mini-video lecture showing you how I decided which solution was the correct one.