11. Probability

 

 

Two events are considered independent if the probability of one event occurring is not changed by whether or not the second event occurred.

 

We write conditional probabilities as P(Event 1 | Event 2).

 

Conditional probabilities tell us the probability of Event 1, given that Event 2 has already happened. If two events are independent, then the two things are unrelated. We calculate conditional probability P( Event 2 | Event 1) by dividing the probability of Event 1 and Event 2 by the Probability of Event 1. The role of conditional probabilities are particularly important when we consider medical screenings.

 

For example, when screening for cervical cancer it used to be recommended that all adult women get screened once a year. But sometimes the results of the screenings are wrong. Either they can say there’s something abnormal when there isn’t (called a false positive) or that everything is all clear when it’s really not (called a false negative). This is exactly the kind of scenario where knowing the likelihood that something is actually abnormal in this case cervical cancer given that you’ve gotten positive tests results would be useful.

 

That is P(Cancer | Positive Test).

 

When looking at the data of people who DON’T have cancer, 3% will get a false positive. And people who DO have cancer will get false negatives 46% of the time.

This means we miss a lot. And maybe freak some people who don’t need to be freaked out.

 

The logic of conditional probabilities can help us make sense of why doctors have recently recommended that these tests be done less frequently in some cases.

In the United States, the rate of cervical cancer is about 0.0081%, so only about 8 in 100,000 women get cervical cancer.

 

Using our rates of false negatives and positives, we can see that for every 100,000 women in the US, only about 4...and we’re rounding here of the about 3,004 people with positive tests actually had abnormal growths.

 

That means the conditional probability of having cancer, given that you got a positive test is only 0.1%. Give or take. And these positive tests require expensive and invasive follow up tests. And I just want to point out that conditional probabilities aren’t reciprocal.

 

That is to say P(Cancer|Pos Test) isn’t the same as P(Pos Test| Cancer) which would be about 50%.

 

https://www.youtube.com/watch?v=OyddY7DlV58

 

 

How do you use probability in your everyday life? Consider things link going out and having fun. What do you need to ensure that you will have a good weekend? And having done these things how likely are you to have a good weekend?