Avoid Duplicate Outcomes

When working through a problem, a typical workflow might be something like:

1. Understand the problem.
2. Create a model of the problem.
3. Perform the calculations for the model.

At step two, you take the scenario and subdivide it into small steps. After that, you figure out what to add and multiply from this model.

Step two is basically figuring out what steps you would take to satisfy the problem, and then figuring out when you should add or multiply based on the model.

Example: Bob flips a coin, and then rolls a die. He records both of the results. How many possible results are there in total?

This can be broken down into two steps:

1. Flip a coin (two outcomes)
2. Roll a die (six outcomes)

From this, we can use the multiplying rule (we don’t need to state that explicitly though, since it’s pretty obvoius). Two steps are independent, and since they are both done, the total number of outcomes is \(2\times 6 = 12\).

Caution

Importantly, make sure:

• your method gives you all the possible outcomes (and doesn’t miss any out), and
• your method avoids duplicate outcomes.

If there are duplicate outcomes, you need to subtract them from the raw answer.

If the model misses some of the possible outcomes, you need to add them back. This is like using the adding rule.

In general, try to avoid duplicate outcomes, unless it is easier to subtract. Make sure you are absolutely certain in what you are doing, and not just doing it because it seems right.

Example: Bob tries to fit four people into a room (for some arbitrary reason), but only two people can be in it at once. The four people consists of two boys and two girls. How many different combination of people can he fit, if he fits two?