oh my god this is the dogs bollocks for my molecular phylogenetics revision!
@Anonymous: Thanks!
@John: Appreciate the reference. Another explanation with a venn diagram: http://blog.oscarbonilla.com/2009/05/visualizing-bayes-theorem/
@Anonymous: Thank you!
@Patty: Glad it helps 
[…] Statistics: Combinations & permutations, Birthday Paradox, Bayes’ Theorem, […]
This is one of the best explanations I’ve found. Perhaps we can see if I really understand it by trying a real world problem I’m wrestling with.
Here’s the data:
- The odds of a chest pain (CP) being caused by a heart attack is 40%.
- The odds of a CP being caused by other factors (anxiety, depression, etc.) is 60%.
- The odds of a heart attack occurring to a female above age 50 is 80%.
- The odds of a heart attack occurring to a female under age 50 is 20%.
I am presented with a 24 year old female who says she is having chest pain. What is the probability that her chest pain is caused by a heart attack? Is it 0.4 x 0.2 = 0.08?
Also, 78% of patients having heart attacks present with diaphoresis (sweating), so 22% of patients having heart attacks don’t sweat. This female is not sweating, so are the odds of her having a heart attack 0.22 x 0.08 = 0.0176?
Thank you!
Thanks for writing this!! Even my stats prof was making this too difficult for everyone, but you have simplified it for me. I now have an understanding of the Bayes formula (enough to write my midterm this morning 
).
thanks! finally got the concept behind bayes rule
Could you work out an example of an email with two words, say ‘Viagra’ and ‘hello’?
[…] An Intuitive (and short) explanation of Bayes Theorem – this is an excellent and concise explanation by Kalid Azad of Better Explained. […]
I didnt get that, bayes theorem is still a tilted pot for me, but thanks for helping!
what would happen if we have to consider other prior probability…lets say, the doctor looked at the symptoms of the patient and guessed that he has 60 percent chance of having cancer. Doctor sends him for the test and test showed positive result. How would we incorporate that 60 percent odd of having cancer based on the patient’s symptoms to the Bayesian equation.
[…] An Intuitive (and Short) Explanation of Bayes’ Theorem Bayes’ theorem was the subject of a detailed article. The essay is good, but over 15,000 words long — here’s the condensed version for Bayesian newcomers like myself. […]
hi
thank you for this article. The first time I came across Bayes Theorem in a business statistics book it was not so clear at all. No it makes more sense for me and its pretty clear…
@France: You’re welcome, glad it helped. I understand it better now too, but there’s still more to go before it’s completely intuitive for me :). I’d like to do a follow-up to this focusing on using the probabilities it predicts.
On the cancer example, it’s interesting to see that a negative test is really significant. That is, if the test says you don’t have cancer, then probability of not having cancer is 99.78% ! So, the value of mammogram is that the healthcare $ can be employed in further investigating the positive (+ and -) cases only.
[…] BetterExplained: An Intuitive (and Short) Explanation of Bayes’ Theorem […]
Thank you for taking the time to write this - it has really helped me get my head around the concept!
@Dan Weisberg: 14% of chest pains in women under 50 are therefore caused by heart attacks. Its (0.20.4)/((0.204)+(0.8*0.6)) = 0.14
@Dan Weisberg: for some odd reason, my posts seem to disappear and reappear… anywho, the other part to your question (as earlier posted) is 4.5% irrespective of when you choose to include the diaphoresis variable.
[…] not completely afraid of math, a great introduction to Bayes’ rule can be found at betterexplained.com. It’s explained from the point of view of somebody using statistics to analyze experiments in […]