How do I differentiate between using regular probabilty and using conditional probablity?

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This question is pretty common in math, actually. In general, regular probabilities tend to be used in only very special cases. For example, if you have multiple independent events or a single event, you will generally be looking at individual probabilities. However, when probabilities start depending on each other or have multiple factors deciding a given probability, you start needing to worry about conditional probabilities.

Conditional probabilities are probabilities that an event occurs given some other condition. For example, the probability that I get into an accident on the freeway depends on whether or not I am tired, drunk, or distracted. These questions often take the form of giving you different probabilities for an event given different conditions.

Similarly, you may see problems with two independent probabilities that are **not mutually exclusive**. In this case, calculating probabilities depends on your determining the expected probability that the two events are occurring simultaneously. For example, if I have a probability of `p_1` that I eat a sandwich at lunch, and a probability of `p_2` that I eat a sandwich at dinner, the probability that I eat a sandwich at neither time is not the following:

`P != 1-(p_1+p_2)`

This equation has the problem that we are double-counting cases because these events are not mutually exclusive! Nothing is stopping me from having a sandwich in both cases, and by considering the two events independently, we are double counting cases where I eat a sandwich for lunch and dinner. So, we need to be a bit clever and use conditional probabilities. For this example, we can start with `p_1`, where I may eat a sandwich at lunch. Then, we determine the **probability that I may have a sandwich at dinner given that I do not eat a sandwich at lunch**.

To find this probability, we first find the probability that I have eaten a sandwich at both meals, which is based on independent probabilities. This is easily given by the following equation:

`p_(1&2) = p_1p_2`

Now, the probability that I have a sandwich at dinner, but not a sandwich at lunch is given the following equation:

`p_(bar(1)&2) = p_2 - p_1p_2`

So our final probability that I do not have a sandwich today is:

`P = 1-(p_1 + (p_2-p_1p_2))`

I suppose these are the main two cases in which I can see you needing to deal with conditional probabilities or converting independent to conditional probabilities. There may be others, but these are the ones we see most often (in math class at least)!

So, a few key phrases that can tell you conditional as opposed to independent? "If-then" statements concerning two probabilities and stems that do not tell you events will be mutually exclusive are big indicators. However, there are always exceptions, so you should watch out for those!

I hope that helps!

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