1.9.1 Intuition
Imagine waking up one morning feeling unwell. As you begin to search the internet to identify the potential cause, you stumble upon a serious ailment, which we'll call “Hypoillness”. Upon reading through the list of symptoms, you discover that you share a striking similarity with the symptoms, approximately 90%. Naturally, this prompts a crucial question: given the high similarity in symptoms, what is the probability that you actually have Hypoillness? However, before assuming a 90% probability of having Hypoillness and panicking, will you try to find the answers to the following questions:
- How prevalent is Hypoillness? In other words, what percentage of the population is affected by this ailment?
- How common are the symptoms you are currently experiencing? i.e., What is the overall percentage of individuals who exhibit these symptoms, whether due to Hypoillness or another cause?
Suppose the response to the first question reveals that Hypoillness is exceedingly rare, affecting just 1 in 10,000 people. Does this revelation sigh relief? Does it make you reconsider the probability of having Hypoillness, given its rarity?
Now, for the second question, what if you discover that the symptoms you're currently experiencing are quite common (such as headaches and coughing), afflicting 8 in every 100 people? Does this newfound knowledge provide a sense of reassurance further?
If your answer to these questions is yes, it's because upon encountering Hypoillness, you didn't solely rely on the fact that you share 90% of the symptoms. Instead, you applied Bayes' theorem to the situation and updated your belief regarding the possibility of having Hypoillness. Assuming that the hypothesis in question is that you have Hypoillness, let's frame all this information within the context of Bayes' Theorem:
- Prior Probability that you have Hypoillness, () = 0.0001 (Since it's extremely rare and 1 in 10,000 people have this illness)
- Probability of you having the symptoms, given you really have Hypoillness, ( | ) = 0.90
- Probability of symptoms in general, () = 0.08 (Since they are quite common and 8 in 100 people have it)
The mathematical formula for Bayes’ theorem is:
Substituting the probabilities into the Bayes’ formula yields:
Even though your symptoms were 90% similar to Hypoillness , the probability of you having it is just 0.1125%!