The Monty Hall problem was created by Steve Selvin and is a classic puzzle whose correct answer is counter-intuitive almost to the point of disbelief. As this page explains, even some of the most competent mathematicians of the 20th century refused to accept the correct answer to the Monty Hall problem for a long time.
Here is the statement of the problem :
Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the other two doors are goats. You pick a door, and the host, who knows what’s behind the doors, opens another door, revealing a goat. He then says to you, “Do you want to change your selection?” Is it to your advantage to switch your choice?
Monty Hall Problem. Image is from wikipedia
What does intuition tell us? After the host opens one door, revealing a goat, we are left with two closed doors, one hiding a car and the other a goat (50% chance of success either way), intuition would lead us to conclude that there is no difference in our chances of success if we switched doors or not.
If only life were that simple………….
In my investigation of Bayes Theorem, I have learned a great many things. Many of these lessons come from listening to the lectures and debates of Dr. Richard Carrier, whose lecture on Bayes Theorem is what got me to first realise the power of Bayes in the first place and how to apply the theorem in a real world situation.
So What is Bayes Theorem?
Bayes Theorem is a mathematical relationship between the probabilities of conditional events. Conditional events are different from independent events in that a conditional event’s chances of occurring are dependent on the chances of occurrence of the event it is conditional upon.
This post is about the application of Conditional Probability and Bayes Theorem in target detection by a randomly maneouvering search platform. Most tutorials about Bayes Theorem on the internet deal with issues like cancer detection maths or picking coloured balls from a hat. I needed a little bit more realism in my quest to understand Bayes Theorem and so I devised the following scenario.
It has taken my small brain months, maybe even years to begin to comprehend Bayes Theorem so don’t be disheartened if you can’t understand this post right away. Keep at it, keep reading, and like me, one day, the truth about our beliefs and the way we form them will dawn on you!!
I have chosen to write this post as a work of fiction rather than a paper on the subject it really addresses. In a nutshell —> Just because your sensor has reported a target detection, it doesn’t mean that there is actually a target there. Continue reading