Bayes' Theorem: A Non-Invasive Look at the Fifth Principle with Our Furry Friends

Having covered the first four principles of Laplace’s probability principles, we’re now ready to explore the fifth principle. In this post, we’ll maintain our feline-themed exploration, using Cool Ranch Dorito, Boggy P, and Garbage Disposal as examples to ensure a fun and engaging learning experience.

Laplace’s Fifth Principle of Probability: Bayes' Theorem

Laplace’s fifth principle introduces Bayes' theorem, which allows us to update the probability of an event based on new evidence. Bayes' theorem can be stated mathematically as P(A|B) = P(B|A) * P(A) / P(B), where P(A|B) represents the probability of event A occurring given that event B has occurred.

Let’s illustrate this principle with an example involving our beloved cats, Cool Ranch Dorito, Boggy P, and Garbage Disposal. Suppose there’s a 60% chance that Cool Ranch Dorito is responsible for knocking over a vase, while Boggy P has a 30% chance, and Garbage Disposal has a 10% chance. You find catnip near the broken vase, and you know that:

• Cool Ranch Dorito loves catnip and will be near it 80% of the time.
• Boggy P is indifferent to catnip and will be near it 50% of the time.
• Garbage Disposal dislikes catnip and will only be near it 20% of the time.

Given this new evidence (finding catnip near the vase), what is the updated probability that Cool Ranch Dorito knocked over the vase?

Applying Bayes' theorem, we first calculate the probability of finding catnip near the vase (P(B)) using the principle of total probability:

P(B) = P(catnip|Cool Ranch Dorito) * P(Cool Ranch Dorito) + P(catnip|Boggy P) * P(Boggy P) + P(catnip|Garbage Disposal) * P(Garbage Disposal) = 0.8 * 0.6 + 0.5 * 0.3 + 0.2 * 0.1 = 0.56

Now, we can calculate the updated probability:

P(Cool Ranch Dorito|catnip) = P(catnip|Cool Ranch Dorito) * P(Cool Ranch Dorito) / P(B) = (0.8 * 0.6) / 0.56 ≈ 0.857

Thus, given the new evidence of catnip near the broken vase, there is approximately an 85.7% chance that Cool Ranch Dorito knocked over the vase.

Up Next: The Sixth and Final Principle of Probability

Stay tuned for our next post, where we’ll complete our journey through Laplace’s principles of probability. We’ll examine the sixth and final principle, providing a comprehensive and entertaining exploration of its significance in modern probability theory and statistical applications.