Evidence under Bayes' theorem
So, let's imagine that you are playing a detective game. You have to find out who ate the last cookie from the jar. You have three suspects: your mommy, your daddy, and your little sister.
You know that your mommy doesn't really like cookies, so she probably didn't eat it. Your daddy loves cookies, but he promised he wouldn't eat any without asking first. And your little sister, well, she's a sneaky one and always loves to eat cookies.
Now, you have some evidence to help you solve the mystery. You found some crumbs on the floor, and it looks like a cookie was eaten in the living room. You also noticed that there was a trail of chocolate stains leading from the living room to your sister's room.
This evidence is what we call "prior probability." It's what we think might have happened before we consider any new information. In this case, the prior probability is that your sister is the most likely suspect because she loves cookies.
But wait, there's more! You remember that your daddy loves chocolate chip cookies, and the trail of stains leads to your sister's room, not the living room. This is what we call "likelihood." It's new information that changes the probability of something happening.
Based on this new information, you might start thinking that maybe your daddy ate the cookie instead of your sister. The evidence of the chocolate stains makes it more likely that your daddy ate the cookie, even though your sister still seems like the most likely suspect based on the prior probability.
This is where Bayes' theorem comes in. It helps you update your probability based on new evidence. It's like adjusting your guess based on new information you find.
Bayes' theorem says that the probability of something happening (like your sister eating the cookie) can change based on new evidence (like the chocolate stains). It takes into account both the prior probability (the likelihood of something happening before we had any evidence) and the likelihood (the new information that changes the probability).
In our cookie detective game, Bayes' theorem can help us calculate the probability of each suspect being the cookie thief. It allows us to weigh the prior probability (how likely they are to eat cookies) with the likelihood (the new evidence we found).
So, in the end, using Bayes' theorem, we might find that even though your sister loves cookies, it's actually more likely that your daddy ate the last cookie because of the chocolate stains leading to his room.
You know that your mommy doesn't really like cookies, so she probably didn't eat it. Your daddy loves cookies, but he promised he wouldn't eat any without asking first. And your little sister, well, she's a sneaky one and always loves to eat cookies.
Now, you have some evidence to help you solve the mystery. You found some crumbs on the floor, and it looks like a cookie was eaten in the living room. You also noticed that there was a trail of chocolate stains leading from the living room to your sister's room.
This evidence is what we call "prior probability." It's what we think might have happened before we consider any new information. In this case, the prior probability is that your sister is the most likely suspect because she loves cookies.
But wait, there's more! You remember that your daddy loves chocolate chip cookies, and the trail of stains leads to your sister's room, not the living room. This is what we call "likelihood." It's new information that changes the probability of something happening.
Based on this new information, you might start thinking that maybe your daddy ate the cookie instead of your sister. The evidence of the chocolate stains makes it more likely that your daddy ate the cookie, even though your sister still seems like the most likely suspect based on the prior probability.
This is where Bayes' theorem comes in. It helps you update your probability based on new evidence. It's like adjusting your guess based on new information you find.
Bayes' theorem says that the probability of something happening (like your sister eating the cookie) can change based on new evidence (like the chocolate stains). It takes into account both the prior probability (the likelihood of something happening before we had any evidence) and the likelihood (the new information that changes the probability).
In our cookie detective game, Bayes' theorem can help us calculate the probability of each suspect being the cookie thief. It allows us to weigh the prior probability (how likely they are to eat cookies) with the likelihood (the new evidence we found).
So, in the end, using Bayes' theorem, we might find that even though your sister loves cookies, it's actually more likely that your daddy ate the last cookie because of the chocolate stains leading to his room.