# Application: Probability Rules

## Particular versus general

First, read the discussion after Example 1.15 of the textbook. In short, the probability that your Powerball ticket is the winning lottery number is extremely small, but if enough tickets are sold then the probability that someone somewhere wins can be high. In general, even if an event has extremely small probability, given enough repetitions of the random phenomenon, the probability that the event occurs on *at least one* of the repetitions is often high. You will construct an example to illustrate this idea.

- Choose something of interest to you that might be considered a “rare event” and find and cite a source that specifies its probability. (For example, the probability that a single ticket wins the Powerball is about 1 in 300 million, but you should
*not*use a lottery example). - Construct your own concrete illustration of how “rare” your rare event is. You are writing your own version of the index card example after Example 1.15 in the context of your problem. Be creative!
- Determine a reasonable context that represents “a large number of
*independent*repetitions”, and compute the*the probability that the rare event happens in at least one of the repetitions*.**Show your work in your calculation**. Explain why it is reasonable to assume the “repetitions” are roughly independent. (For example, if 500 million Powerball tickets are sold, the probability that there is at least one winning ticket is 0.8. See Example 3.16 for examples of similar calculations.) - Write a few sentences describing in your own words how your example illustrates the idea of “the particular versus the general”. Your explanation should be non-technical and understandable to someone with no background in probability and statistics.

## Create your own problems

Create and solve your own problems using some of the techniques from Chapter 11, Chapter 12, and Chapter 13. You should

- Describe your problem set up and context, and specify the questions/problems to solve
- Describe why you’re interested in these problems and context.
- Solve the problems, showing all your work
- Interpret your solutions in context

Collectively, your problems should involve application of all of the following:

- Multiplication rule for non-independent events
- Multiplication rule for independent events
- Law of total probability
- Bayes rule
- Counting rules

You can choose if you want a single problem set up/context with multiple parts, or multiple problems with different set ups/contexts. However you structure your collection of problems, you should apply each of the 5 strategies above at least once.

Choose contexts and applications that are interesting to you personally. Your problems should be as realistic as possible within your contexts. For example, if you specify the probability of an event in your set up, briefly justify why this is a reasonable value in context (possibly by providing a reference to a source).

The problems do not necessarily need to be more difficult than the ones we have seen in class. You should be creative in your choice of contexts, and each problem should be interesting and relevant in context, but it’s fine if the method of solving the problem is similar to another problem we have seen. However, you are strongly encouraged to be creative!

Solve the problems by hand, detailing all your steps as if you are teaching someone how to solve the problem. You should rely on properties of probability and distributions as much as possible, and only use calculus/algebra when absolutely necessary. You can use software like WolframAlpha to do any calculus/algebra, but make sure you clearly set up the equations to solve or expressions to evaluate. You can assume the person you are teaching has seen all the topics and concepts that we have; they just haven’t seen your problems before.

The problems that you construct should be your own creation, and your solutions should be your own work. You are NOT allowed to copy material from outside sources. You can consult sources, but you must CITE any sources that you consult and include references (e.g., to justify a numerical value of a probability in context). In particular, you can NOT just copy an exercise or its solution from a textbook or other online source. To repeat: The problems that you construct should be your own creation, and your solutions should be your own work.