Bayes and The Bachelor
Breaking down Bayes’ problems with The Bachelor… because what’s more romantic than love and statistics?
Let’s be honest, at the end of a Monday the last thing you usually want to be doing is math. You would probably much rather be doing something relaxing like reading a book, cooking a nice meal, or if you’re like me… watching The Bachelor. But what if you have a statistics test later that week and suddenly you feel guilty about taking the time to see who the bachelor is going to fall in love with. Well, fear not… cause we can use The Bachelor to practice our statistics!
In this blog I will be breaking down how to solve a Bayes’ problem with the tree method by calculating what a bachelor contestant’s chances of having received the first impression rose is given they made it to the final four.
Bachelor Basics
First things first, let us start with some Bachelor basics for those lucky souls who have yet to become addicted to The Bachelor franchise. The show is about a bachelor courting contestants to find his true love. Every week there is a limited amount of roses for him to hand out and if you receive a rose you get to continue vying for his heart for another week. The season begins with the bachelor courting anywhere between 25 to 32 contestants. For 18 seasons of The Bachelor, on the first night of the season the bachelor hands out a single coveted first impression rose to a contestant he feels the most “sparks” with… or really just a contestant he definitely wants to keep around for another week. Alright, now that we got that down let’s review some Bayes.
Bayes’ Theorem Review
Bayes can be used to predict the probability of an event occurring based on prior knowledge of conditions that may be related to the event with the equation :
P(A|B): our posterior, is the probability A happens given that B occurred
P(B|A): our likelihood, is the probability B happens given that A occurred
P(A): our prior, is the independent probability of A occurring
p(B): our marginal likelihood, is the independent probability of B occurring
While this does appear pretty simple to solve, a simple plug and chug, I find the challenge in Bayes’ problems is finding the correct numbers for the variables. One easy way to find the variables is to use a tree method. What’s that you say? Well let’s break it down!
Using The Tree Method To Solve The Bachelor Bayes’ Problem
(Disclaimer- these numbers are not 100% accurate due to rounding)
Bayes’ Bachelor Practice Problem:
I’m thinking about going on The Bachelor… okay not really, don’t think my husband would approve. But for the sake of this problem let’s pretend that I want to go on The Bachelor. I know that I only have a 3.65% chance of receiving the first impression on the first night. I also know that if I receive that first impression rose, I have a 55.56% chance of making it to the final four hometown dates. However, if I don’t get the first impression rose I know I only have a 13.05% chance of making it to the final four. I want to find out what the probability is that I had received the first impression rose given I made it to the final four?
To start, we know that our P(A) is our probability of getting the first impression rose P(FR) and P(B) is the probability of making it all the way to the final four P(FF). Let’s setup our equation using our Bachelor variables.
Now let’s breakdown how to use the tree method to find all our variables. The top row represents a general layout for each step, while the bottom row represents each step applied to our Bachelor problem.
Wow! That wasn’t too hard to make. Now that we know how to make our tree, how do we read our trees? I like to think of it as reading from right to left. Let’s look at the pictures below to help us out. We start by looking at what condition we want B to fulfill (the red circle), given what condition we want A to fulfill (red arrow).
Now that we understand how to read our trees, let’s finish solving our problem! Before we can do this we need to know that every two breaking points need to add up to the total one.
Okay, now we can continue on to step 4:
Hooray! Thanks to the tree we now know all the variables we need to solve our equation. Let’s plug them in below to find our answer.
Not only did we get to breakdown the tree method of solving a Bayes’ problem, but now we also know that if we ever decide to go on The Bachelor and make it to the final four hometown dates, there is a 13.89% chance we had received the first impression rose.
