Hello! Today we'll learn about Bayes' formula through a fun example with colored balls and bags. Imagine you have two bags with different colored balls inside. Bayes' formula helps us figure out which bag a ball came from when we only know its color!
Now let's set up our problem clearly. We have Bag A with 1 red ball and 1 blue ball. We also have Bag B with 2 red balls and 1 blue ball. We will pick one bag randomly, then draw a ball from it. Our question is: if we draw a red ball, which bag did it most likely come from?
Let's understand the basic probabilities. Since we pick a bag randomly, the probability of choosing Bag A is one half, and the probability of choosing Bag B is also one half. If we pick from Bag A, the chance of getting a red ball is one out of two, or one half. If we pick from Bag B, the chance of getting a red ball is two out of three, since there are 2 red balls and 1 blue ball.
Now let's apply Bayes' formula! First, we calculate the total probability of getting a red ball from either bag. This equals one half times one half plus two thirds times one half, which gives us seven twelfths. Then we can find that if we drew a red ball, there's a three sevenths chance it came from Bag A, and a four sevenths chance it came from Bag B. So the red ball most likely came from Bag B!