91Ó°ÊÓ

When we talk about joint probability, we are referring to the probability of two events happening at the same time. In this context, we're interested in both event B and an event from A happening together. To compute joint probabilities, we use the formula:

• \( P(B \cap A_i) = P(B | A_i) \cdot P(A_i) \)

This formula is quite straightforward. It simply tells us to multiply the conditional probability of B given \(A_i\) by the probability of \(A_i\). For example:

• \( P(B \cap A_1) = 0.5 \times 0.2 = 0.1 \)
• \( P(B \cap A_2) = 0.4 \times 0.5 = 0.2 \)
• \( P(B \cap A_3) = 0.3 \times 0.3 = 0.09 \)

These calculations allow us to understand the likelihood of the combination of events B and each \(A_i\) occurring in our scenario.

Conditional probability is a measure of the probability of an event occurring given that another event has already occurred. In simpler terms, it helps us adjust our probabilities based on additional information, showing us how probabilities alter under certain conditions.

It is represented as \( P(B | A) \), read as 'the probability of B given A.' The formula is:

\[ P(B | A) = \frac{P(B \cap A)}{P(A)} \]
In our exercise, this concept is specially used to link the occurrence of events B and each event A. The conditional probabilities were used in computing the joint probabilities initially. This is because, in those calculations, the condition that A had occurred was already considered, simplifying our calculations. By understanding conditional probability, we can make precise predictions in probability-related situations.

Posterior probability is an update of the initial probability of an event after taking into account new evidence or information. It reflects what we know about an event’s probability after considering what has happened. This concept is a critical part of Bayes’ Theorem.

Bayes’ Theorem helps us quantify the posterior probability by inversely weighting the initial, or prior, probabilities with the likelihood of the observed evidence. The formula is:

\[ P(A_i | B) = \frac{P(B \cap A_i)}{P(B)} \]
In our problem, we used this formula to find the updated probabilities of events \(A_1\), \(A_2\), and \(A_3\) given that event B occurred. For example, the posterior probability of \(A_2\) given B, represented as \(P(A_2 | B)\), was calculated as:

• \(P(A_2 | B) = \frac{0.2}{0.39} \approx 0.513 \)

Bayes' theorem thus helps us transition from a world of initial assumptions to more informed decisions, integrating fresh evidence into our probabilistic models.