How to label the events in a probability problem?

Question

I'm starting a statistics and probability class and it's still pretty hard for me to "name" my probability events, which I think makes it very difficult to solve problems.

I'll pick an example from my exercises to illustrate my problem :

We asked 1000 persons which magazines do they regularly read between magazine A, B and C. We obtained the following results : 60% read A, 50% read B, 50% read C. There are 20% which read B and C, 30% read A and C and 30% read A and B. There is 10% of the 1000 persons who read the 3 magazines. Given one person, calculate the probability that this person:

1. Reads A or C.
2. Doesn't read a magazine at all
3. Reads A but not B
4. Reads only one magazine.

So, in this example the events were labelled :

A = {Probability that this person reads A}

B = {Probability that this person reads B}

C = {Probability that this person reads C}

If this were an exercise, I might have started by naming the probability event according to the question so :

X = {Probability that this person read's A or C}

Y = {Probability that this person doesn't read a magazine}

etc..

or something like that. I would have messed around for a while, not being able to find the answer, and at some point I would have figured out the events like they are labelled in the example.

I know that this example is pretty easy to figure out, but in more complex exercices I find it difficult to label the events properly. IMO, it's kind of like not being able to identify variables properly in an algebra problem. Without proper variables it's impossible to figure out the answer.

So, I'm sure there's a "way to follow" in order to label the events in a probability problem and I would like to find out what it is.

Edit

For my current problem, there are three cards. One with a red face, red background. The other with a red face, black background and the last with black face black background.

I need to find the probability that given a picked card where the face is red, the background is black.

I am not looking for answers to the problem

I'd just like to know how I'm supposed to figure out what should be the labels to my events. One per card? One per face color?

Answer 1

We have some core concepts that you must understand:

Discrete Probability Theory:
Outcome \ $\omega$
Sample Space \ $\Omega$
Event \ $A$
Experiment \ (how to model a word problem)

and

${\displaystyle \qquad P(A)=\sum _{\omega \in A}p(\omega )\quad {\text{for all }}A\subseteq \Omega }$

Also, you can set up a model that has more elementary outcomes than you are interested in - you want to 'collapse' this easily described model by insisting that some event is occurring.

With complex problems, the experiment can be set up in more than one way; there is no 'cookie cutter' solution system. So you have to think things out. Instead of jumping right into a problem, there is nothing wrong with thinking first about what you can easily model and is in the 'ballpark'.

It is always good to model things with simple outcomes. For your problem, we can consider an experiment $\mathcal E$ as follows:

Select a card and record the color of the face and background as an ordered pair, so that

$\tag 1 \Omega = \{(R,R), (R,B), (B,B)\}$

Of course for our mind experiment $\mathcal E$ each of these outcome has a probability of $1/3$.

Do you want to model your exact problem with a new experiment? Or perhaps you can use some conditional probability formulas. Can you guess the answer? If you can guess the answer can you verbally justify it?

The one thing you have to remember is that the probabilities of the outcomes always add up to $1$. So if you are ruling out some outcomes by conditioning, the remaining outcomes must be 'pumped up' in some 'fair' way.

Note: Nobody can stop you if you don't label your events and simply list them out as subsets of $\Omega$.

Answer 2

The right definition is the one that allows you to prove the right theorem! Ultimately the issue at hand is not quite one of naming, but rather one of choosing which "parts" of a problem to focus on. This is an art, crucial to all branches of knowledge (not just of probability or mathematics), without hard and fast rules.

For probability, I tend to find the following piece of advice useful. It's easier to start with lots of "elementary events", and later "condense" them into more complex events, than the other way around. This is particularly useful if one manages to break the probability space into independent or mutually exclusive events (or, failing that, non-positively correlated events), because 1) there are some very strong tools one can draw upon in those cases and 2) those cases are generally easier to reason about without making mistakes.

So, in your example, the suggestion would boil down to "one event per card" (because these are mutually exclusive events)!

Answer 3

In some sense, I would like to elaborate on previous answers and say that "less is more" and you should keep it simple:

As you observed, having too many variables is hard to keep track of, unless you have some way of relating these symbols together (maybe using a diagram?). So my answer is: use as few symbols as you can get away with!

E.g. In your first problem you started perfectly. Once you defined $A$, $B$, and $C$ in the way you did, you can express all the other events ($X$, $Y$ and so on) in terms of these three (using or's, not's & and's), without needing to give them any names of their own. If absolutely necessary, you can quickly write down the event in shorthand. Anything is good as long as it is meaningful to you (and secondly, the reader).

Your marker understands the context of the question well enough that this is okay! It also saves you time, and gives you less distraction.

I.e. your answer to each of the sub-questions may begin:

$$ \begin{align} 1. &\ \mathbb P (A \text{ or } B) = \cdots \\ 2. &\ \mathbb P (\text{[not }A]\text{ and [not }B]\text{ and [not }C]) = \cdots \\ 3. &\ \mathbb P (A\text{ and [not }B]) = \cdots \\ 4. &\ \mathbb P (\textit{reads only one}) = \mathbb P ([A\text{ and not }B\text{ and not } C]\text{ or }\cdots \end{align} $$ and so on.

E.g. Answering your second question, I would begin:

• Let $RR$ denote the event that I select the red face, red background card.
• Let $RB$ denote the event that I select the red face, black background card.
• Let $BB$ denote the event that I select the black face, black background card.

Then $\mathbb P (\textit{red face}) = \mathbb P( RB \text{ or } RR)\ \cdots\ $

and so on.

These are all the "variables" you need!

To finish the question from that point on is simply a case of recognising how the events given to you relate to the events you want to calculate (i.e., the event that the card chosen has a black background, given that it has a red face) and reduce it to an arithmetic problem.

There are perhaps two ways of approaching this:

1. Symbolically: if you're happy with formulas, once you've written an expression for the event in terms of $A$'s $B$'s and $C$'s say, you simply apply the formulas you're comfortable with.
2. Visually: you use a tree diagram or a Venn Diagram (or perhaps a Carell diagram!) to identify the events that correspond to your required answer.

But perhaps this is outside the scope of the question.