How to Calculate Revised Probabilities of Events with Extra Information

With the advancement of technology, we have access to more data than ever before. However, this additional information can sometimes create uncertainty and confusion when trying to calculate probabilities of an event. In this article, we will explore the concept of how to calculate revised probabilities of events with extra information.

Before delving into the topic, it’s essential to understand what probability is. Probability is a mathematical tool that helps us determine the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 means that the event is impossible, and 1 means that the event will definitely happen.

Let’s take an example of tossing a coin. The probability of getting heads is 0.5, and the probability of getting tails is also 0.5. This means that we have an equal chance of getting either outcome when we toss a coin.

Now, let’s consider a scenario where we have a bag of marbles. This bag contains 20 marbles, of which 5 are red and 15 are blue. The probability of picking a red marble randomly from the bag is 5/20 or 0.25. Similarly, the probability of picking a blue marble is 15/20 or 0.75.

However, what happens when we have extra information that could impact the probabilities? For instance, suppose we know that five of the blue marbles are defective, and one of the red marbles is defective. In that case, the probabilities change, and we need to recalculate them based on the new information.

To calculate revised probabilities in such situations, we can use conditional probability. Conditional probability is a probability that takes into account additional information, given that we already have some knowledge of the event.

To calculate conditional probabilities, we use the formula:

P(A|B) = P(A∩B)/P(B)

Where P(A|B) is the probability of event A given that event B has occurred, P(A∩B) is the probability of events A and B occurring together, and P(B) is the probability of event B occurring.

Let’s apply this formula to our bag of marbles example with extra information. Suppose we want to find the probability of picking a non-defective blue marble from the bag. We already know that there are five defective blue marbles in the bag. Hence, the probability of picking a non-defective blue marble is:

P(non-defective blue) = P(blue | non-defective) * P(non-defective)

P(blue | non-defective) = the probability of picking a non-defective blue given we are picking a blue marble = (15-5)/15 = 0.67

P(non-defective) = the probability of picking a non-defective marble = (15-5+1)/(20-1) = 11/19

Hence, P(non-defective blue) = 0.67 * 11/19 = 0.39

In summary, when we have extra information that can affect the probabilities of an event, we need to recalculate the probabilities using conditional probability. While this may seem complex, it is a crucial tool for decision-making, especially in industries that rely heavily on probability calculations, such as insurance, finance, and sports.

In conclusion, probability is an essential concept in decision-making, and as we’ve seen in this article, it’s critical to always recalculate probabilities when new information becomes available. By using conditional probability, we can revise the probabilities of events and make informed decisions based on the new information.