In the above question, we are discussing conditional probability and marginal probability.
Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted by P(A∣B), which is read as “the probability of event A given event B.” The formula for conditional probability is:
In simpler terms, conditional probability reflects the updated probability of event A occurring, taking into consideration that event B has already occurred. It adjusts the probability based on the information or occurrence of another event.
For example, if you have two events, A and B, the probability of A occurring might be different when you know that B has occurred compared to when you don’t have that information. Conditional probability helps quantify this change in probability based on the occurrence of another event.
Marginal probability refers to the probability of a single event or variable without considering the occurrence or non-occurrence of other events. It is essentially the probability of a single event, irrespective of the outcomes of other events. The term “marginal” is used because it is often presented in the margins of probability tables.
For a single event A, the marginal probability is denoted as P(A). The formula for marginal probability is straightforward:
where the sum is taken over all possible events Bi that could occur. In simpler terms, marginal probability represents the overall likelihood of event A happening, regardless of the different conditions or scenarios.
To illustrate, consider the following example with two events A and B:
- P(A): The probability of event A occurring.
- P(B): The probability of event B occurring.
- P(A∩B): The joint probability of events A and B (the probability that both A and B occur).
The marginal probability of A is simply P(A), and it is not influenced by the occurrence or non-occurrence of event B. It provides a baseline probability for the individual event.
Note: The concepts related to the question is discussed in this blog for reference purposes of the students only.
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