Reading The “Given That” Versus

The “AND” StatementBy Henry Mesa

Many students struggle with deciphering when a written statement translates to the function P(A | B) and when it translates to the function P(A AND B).

So this presentation attempts to provide you with some clarification.

Suppose that in a population 10% of the people smoke, also in that same population 0.1% of the people have lung cancer. Of the smokers, 27% have lung cancer.

Here is the translation of the first two, P(smoke) = 0.1 P(lung cancer) = 0.001.

Of the smokers, 27% have lung cancer. How do we translate this statement?

P(lung cancer | smoke) = 0.27, that is we are talking about those with lung cancer but only if the person is a smoker, “Of the smokers… “ a given.

How many other ways could I have written the same statement?

Suppose that in a population 10% of the people smoke, also in that same population 0.1% of the people have lung cancer. Of the smokers, 27% have lung cancer.

Here is the translation of the first two, P(smoke) = 0.1 P(lung cancer) = 0.001.

Given that we are only considering smokers, 27% of them have lung cancer.

Or

Let a person be a smoker, the chance they have lung cancer is 27%.

Or

If a person is a smoker, the chance that they have lung cancer is 27%.

Or

Suppose that a person is a smoker, the chance that they have lung cancer is 27%.

Or

It turns out that 27% of the smokers have lung cancer.

Compare this to the following statement

Suppose that in a population 10% of the people smoke, also in that same population 0.1% of the people have lung cancer. It turns out that 0.01% of the population are smokers with lung cancer.

This translates to P(smoke AND lung cancer). Why? “It turns out that 0.01% of the population …” the 0.01% only refers to the population that is the “whole” (sample space) we are considering. Unlike, the previous statements.

Given that we are only considering smokers, 27% of them have lung cancer.

Or

Let a person be a smoker, the chance they have lung cancer is 27%.

Or

If a person is a smoker, the chance that they have lung cancer is 27%.

Or

Suppose that a person is a smoker, the chance that they have lung cancer is 27%.

Or

It turns out that 27% of the smokers have lung cancer.

Keep in mind that in order to translate into a “given that” P(B |A) you need to have other probabilities in order to “signal” that you have changed the sample space. That is what the “|” indicates, a change in sample space.

P(B | A) “what is the probability that B occurs given that we have changed the sample space from its original instead be defined by event A.” So this means there needs to be the statement P(B) present somewhere.

A particular product has 3% chance of not working properly. For that product, 7% of the time the product gets returned to the store. Lastly, for that product, 1% of the time that product is returned and also it does not work. Translate all three probabilities using function notation.

P(not work) = 0.03 P(return) = 0.07, “For 1% of the products…”

P(not work AND return) = 0.01

A particular product has 3% chance of not working properly. For that product, 7% of the time the product gets returned to the store. Of the items returned, 14.29% of them do not work properly. Translate the last statement.

“Of the items returned

P(not work | return) = 0.1429

A particular product has 3% chance of not working properly. For that product, 7% of the time the product gets returned to the store. If an item is returned, 14.29% of the time it is due to the item not working properly. Translate last statement.

“If an item is returned…

P(not work | return) = 0.1429

A particular product has 3% chance of not working properly. For that product, 7% of the time the product gets returned to the store. If a person finds that the item does not work properly, there is a 33.33% chance that they will return it. Translate last statement

“If a person finds that the item does not work properly

P(return | not work) = 0.3333

Notice that in this case we know the item does not work properly, we are merely stating the chance of it being returned given that we know it does not work.

I hope this has helped.