Probability Formulas, Symbols & Notations - Marginal, Joint, & Conditional Probabilities
The Organic Chemistry Tutor・19 minutes read
Marginal probability calculates the probability of a single event occurring independently, while conditional probability determines the likelihood of an event happening given that another event has already occurred. Understanding these concepts is essential for differentiating between events occurring together simultaneously or in a specific order.
Insights
- Marginal probability is the likelihood of a single event happening on its own, without considering other events, calculated by dividing successful outcomes by total possible outcomes.
- Conditional probability involves the likelihood of an event occurring given that another event has already happened, determined by the probability of A given B times the probability of B. This concept is essential in distinguishing between events happening together simultaneously or in a specific order.
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Recent questions
What is marginal probability?
Marginal probability is the probability of a single event occurring independently of other events, calculated as the number of successful outcomes divided by the total possible outcomes.
How is conditional probability calculated?
Conditional probability is the probability of an event occurring given that another event has already occurred, calculated as the probability of A given B times the probability of B.
What is the difference between independent and dependent events?
Independent events have the probability of A given B equal to the probability of A, while dependent events are where the outcome of one event affects the probability of the other event.
What is the formula for Bayes' theorem?
Bayes' theorem is calculated as the probability of A given B equals the probability of B given A times the probability of A, divided by the probability of B.
How is negation probability calculated?
Negation probability, or the complement of an event, is calculated by subtracting the probability of the event from 1. This concept is crucial in understanding probabilities within a sample space.
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Summary
00:00
Probability Fundamentals: Union, Intersection, and Conditional
- Marginal probability is the probability of a single event occurring independently of other events, calculated as the number of successful outcomes divided by the total possible outcomes.
- Symbols for Union and Intersection in probability are represented as ∪ and ∩ respectively.
- Union probability is the probability that either event A or event B will occur, calculated using the addition rule.
- Mutually exclusive events have a joint probability of zero as they do not intersect.
- Joint probability is the probability that both events A and B will occur simultaneously, calculated using the multiplication rule.
- For independent events, the probability of A given B is equal to the probability of A, and the probability of B given A is equal to the probability of B.
- Dependent events are those where the outcome of one event affects the probability of the other event, unlike independent events.
- The probability of events occurring together can be represented as P(A ∩ B) or P(B ∩ A), depending on the order of occurrence.
- Conditional probability is the probability of an event occurring given that another event has already occurred, calculated as the probability of A given B times the probability of B.
- Understanding the context of a probability problem is crucial to differentiate between events occurring together simultaneously or in a specific order.
23:45
Understanding Conditional, Bayes' Theorem, and Negation Probability
- Conditional probability is the likelihood of event A happening given that event B has already occurred. It is calculated by dividing the joint probability of A and B by the marginal probability of B, or by using the formula P(B|A) = P(A and B) / P(A).
- Bayes' theorem is another method to calculate conditional probability, where the probability of A given B is equal to the probability of B given A times the probability of A, divided by the probability of B. This formula is related to the joint probability rule and can be compared to the standard conditional probability formula.
- Negation probability, or the complement of an event, is calculated by subtracting the probability of the event from 1. For example, if the probability of event B occurring is 0.35, the probability of B not occurring is 0.65. This concept is crucial in understanding probabilities within a sample space.
