Counterintuitive Probabilities Suggested by Bayes’ Theorum

If you take a medical test that is 99% accurate and you get a positive test result back, what do you think is the probability that you actually have the disease? If you think it’s 99%, it’s not… it’s actually 9 percent.

Actual probabilities of an event and assumed probabilities of events are two very different things. It is this conclusion that Veritasium looks at in their most recent video.

According to Edward Arnold, author of Kendall’s Advanced Theory of Statistics, Bayes’ theorem is stated as this equation:

{\displaystyle P(A\mid B)={\frac {P(B\mid A)\,P(A)}{P(B)}},}

where A and B are events and P(B) ≠ 0.

  • P(A) and P(B) are the probabilities of observing A and B without regard to each other.
  • P(A | B), a conditional probability, is the probability of observing event A given that B is true.
  • P(B | A) is the probability of observing event B given that A is true.

Understanding this theorem may prove tricky at first, but it has significant implications for daily life once grasped. Essentially, the probability of something happening or being correct is dependent on the probability of the event occurring prior to its occurrence. So, for the case of a medical test. Your actual probability of having a disease is not only dependent on the accuracy of the test, but also on the probability of you having the disease. Thoroughly confused? Check out the intriguing video from Veritasium below.

Trevor is a civil engineer by trade and an accomplished internet blogger with a passion for inspiring everyone with new and exciting technologies. He is also a published children’s book author whose most recent book, ZOOM Go the Vehicles, is aimed at inspiring young kids to have an interest in engineering.


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