Conditional Probability: Unboxing the Third Principle with Our Feline Friends

Having covered Laplace’s first and second principles of probability, we’re ready to dive into the third principle. In this post, we’ll continue our feline-themed exploration, using Cool Ranch Dorito, Boggy P, and Garbage Disposal as examples to make the learning experience enjoyable and engaging.

Laplace’s Third Principle of Probability: Conditional Probability

Laplace’s third principle deals with conditional probability, which is the probability of an event occurring given that another event has already occurred. The principle can be stated mathematically as P(A|B) = P(A and B) / 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 you have three boxes, each containing one of the cats. You know that:

• Box 1 has a 50% chance of containing Cool Ranch Dorito,
• Box 2 has a 30% chance of containing Boggy P, and
• Box 3 has a 20% chance of containing Garbage Disposal.

Now, let’s say you’re told that one of the boxes contains a cat wearing a red collar, and you know that:

• Cool Ranch Dorito has a 70% chance of wearing a red collar,
• Boggy P has a 40% chance of wearing a red collar, and
• Garbage Disposal has a 90% chance of wearing a red collar.

Given this information, what is the probability that Box 1, which has a 50% chance of containing Cool Ranch Dorito, also contains a cat wearing a red collar?

Applying Laplace’s third principle, we first determine the probability of the compound event (Box 1 containing Cool Ranch Dorito and the cat wearing a red collar):

P(Box 1 contains Cool Ranch Dorito and red collar) = P(Box 1 contains Cool Ranch Dorito) * P(red collar|Box 1 contains Cool Ranch Dorito) = 0.5 * 0.7 = 0.35

Next, we need to find the probability of any box containing a cat with a red collar:

P(red collar) = P(Box 1 contains Cool Ranch Dorito and red collar) + P(Box 2 contains Boggy P and red collar) + P(Box 3 contains Garbage Disposal and red collar) = 0.5 * 0.7 + 0.3 * 0.4 + 0.2 * 0.9 = 0.35 + 0.12 + 0.18 = 0.65

Now, we can calculate the conditional probability:

P(Box 1 contains Cool Ranch Dorito|red collar) = P(Box 1 contains Cool Ranch Dorito and red collar) / P(red collar) = 0.35 / 0.65 ≈ 0.538

Thus, given that a cat is wearing a red collar, there is approximately a 53.8% chance that Box 1 contains Cool Ranch Dorito.

Up Next: The Fourth Principle of Probability

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