Probability Formulas, Symbols & Notations - Marginal, Joint, & Conditional Probabilities

The Organic Chemistry Tutor

24 Sept 202330:43

EducationalLearning

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TLDRThe video provides key probability formulas and concepts for students learning the topic. It covers calculating marginal, union, joint, conditional, and negation probabilities. It explains the symbols for union and intersection and how to tell if events are independent, dependent, or mutually exclusive. It discusses the addition rule for 'or' and multiplication rule for 'and.' It also compares the conditional probability and Bayes Theorem formulas. Overall, the video aims to equip students with the main formulas and conceptual understanding needed to solve common probability problems.

Takeaways

😀 Marginal probability calculates the chance of a single event occurring independently.
📊 Union probability is the chance that either event A or event B occurs.
🔀 Joint probability is the chance that both events A and B occur together.
❗️ Conditional probability calculates the chance of A given B has already occurred.
✅ Independent events do not affect each other's probabilities.
🔗 Dependent events have probabilities that rely on prior events.
☑️ The addition rule applies when calculating 'A or B'.
✖️ The multiplication rule applies when calculating 'A and B'.
🔻 Bayes' theorem relates conditional and joint probability.
❌ Negation probability is the chance that an event does NOT occur.

Q & A

What is marginal probability?
-Marginal probability is the probability of a single event occurring independent of other events. For example, what is the probability of event A occurring given a sample space of numbers 1-9.
What do the symbols ∪ and ∩ represent?
-The symbol ∪ represents union, combining the elements of two sets. The symbol ∩ represents intersection, finding the common elements between two sets.
How do you calculate union probability?
-Union probability is calculated using the addition rule: P(A or B) = P(A) + P(B) - P(A and B). For mutually exclusive events, P(A and B) = 0.
What is joint probability?
-Joint probability is the probability that both events A and B occur together simultaneously. It's calculated as P(A and B) = P(B|A)P(A) = P(A|B)P(B).
What are some examples of independent vs dependent events?
-Tossing coins and drawing marbles with replacement are independent events. Drawing marbles without replacement are dependent events since the probabilities change based on previous draws.
What does P(A|B) represent?
-P(A|B) represents the conditional probability - the probability of A occurring given that B has already occurred.
How are conditional probability and Bayes theorem related?
-Conditional probability P(A|B) = P(A & B) / P(B). Bayes theorem gives an equivalent form using P(B|A) instead.
When can P(AB) and P(BA) be different?
-P(AB) and P(BA) represent the order of events, with A occurring before B vs B before A. So they can be different if the order matters.
What is negation probability?
-Negation probability is the probability that the complement of an event occurs, equal to 1 - P(A). For example, if P(B) = 0.35, P(not B) = 1 - 0.35 = 0.65.
What key formulas should I memorize for probability?
-Key formulas: marginal probability, addition/multiplication rules, conditional probability, Bayes theorem, joint probability, negation probability. Understand when to apply each one.

Outlines

00:00

😀 Introducing Key Probability Concepts and Formulas

The first paragraph introduces key concepts in probability that will be covered in the video, including types of probabilities, symbols and notation, independent and dependent events, mutually exclusive events, and formulas that will be discussed.

05:00

😃 Understanding Marginal, Union, and Intersection Probability

The second paragraph explains marginal, union, and intersection probability using a specific example and numerical sample space. It covers calculating probability for individual events as well as combined events using addition and multiplication rules.

10:01

😄 Distinguishing between Mutually Exclusive and Non-Mutually Exclusive Events

The third paragraph explains the difference between mutually exclusive and non-mutually exclusive events, using Venn diagrams to visualize the concepts. It also shows how to calculate joint and union probability for these different types of events.

15:02

😁 Calculating Joint, Conditional, and Bayes Theorem Probability

The fourth paragraph covers formulas for calculating joint probability, conditional probability, and Bayes theorem. It explains the multiplication rule for joint probability and the relationship between conditional probability and Bayes theorem.

20:02

😆 Independent vs. Dependent Probability Events

The fifth paragraph distinguishes between independent and dependent probability events using examples of drawing marbles and tossing coins. It explains how replacement affects dependency between sequential events.

25:03

🤔 Clarifying Order and Simultaneity in Probability Expressions

The sixth paragraph notes that probability expressions may or may not imply order of events, clarifying the potential ambiguity. It recommends specifics like subscripts to denote clear event sequence.

30:04

😯 Introducing Conditional Probability and Bayes Theorem

The seventh paragraph formally introduces the concepts of conditional probability and Bayes theorem. It relates them through joint probability and marginal probability formulas.

🥳 Covering Additional Key Probability Concepts

The final paragraph notes coverage of key probability concepts and formulas, inviting viewers to explore additional example problems in linked materials.

Mindmap

Keywords

💡Marginal probability

Marginal probability refers to the probability of a single event occurring independent of other events. For example, in the video the probability of getting outcomes 1, 2, 3 or 4 (event A) is calculated independently without considering the probabilities of other events B or C. Marginal probability is a building block for calculating more complex conditional and joint probabilities.

💡Union probability

Union probability is the probability that either event A or event B occurs, or that both events occur. It encompasses all outcomes where A or B happen. For independent events, union probability is calculated by simply adding the individual marginal probabilities. The video shows how to calculate P(A or B) using the outcomes included in A union B.

💡Joint probability

Joint probability refers to the probability that both events A and B occur together simultaneously. It is calculated by multiplying the probability of A given B has occurred by the probability of B. For independent events, it is simply the product of the individual marginal probabilities P(A) and P(B).

💡Conditional probability

Conditional probability is the probability of one event A occurring given that another event B has already happened. It is calculated by dividing the joint probability P(A and B) by the marginal probability P(B). Bayes' theorem provides an equivalent calculation in terms of P(B given A) instead.

💡Independent events

Independent events are those whose probabilities do not depend on each other. For independent events, the conditional probability P(A given B) simply equals the marginal probability P(A). The video gives examples like drawing marbles with replacement and coin tosses.

💡Dependent events

Dependent events are those where the probability of one event relies on or is affected by the other event. Drawing marbles without replacement makes the probability of each subsequent draw dependent on the prior draws.

💡Mutually exclusive

Mutually exclusive events are those that cannot occur simultaneously, such that P(A and B) = 0. The video shows events A and B as mutually exclusive since they do not share any elements in their sample spaces.

💡Complementary event

The complement of an event A consists of all outcomes in the sample space that are not included in A itself. Negation or complementary probability P(not A) is calculated by subtracting the probability P(A) from 1.

💡Multiplication rule

The multiplication rule of probability states that for two events A and B, the probability P(A and B) = P(B given A) x P(A) = P(A given B) x P(B). This allows calculation of joint probabilities.

💡Addition rule

The addition rule of probability is used when considering disjunctions/unions of events. For events A and B, P(A or B) = P(A) + P(B) - P(A and B). The last term avoids double counting outcomes in the intersection.

Highlights

Marginal probability is the probability of a single event occurring independent of other events.

The probability that an event will occur is equal to the number of successful outcomes divided by the total possible outcomes.

Union probability is the probability that either event A will occur, or event B will occur, or both event A and B will occur.

The addition rule: The probability that event A or event B will occur is equal to the probability of A plus the probability of B minus the probability of A and B occurring together.

Joint probability is the probability that both events A and B will occur together simultaneously at the same time.

For independent events, the joint probability is simply the probability of A times the probability of B.

Conditional probability is the probability of event A occurring given that event B has already occurred.

Conditional probability equals the joint probability divided by the marginal probability.

Bayes' theorem gives another way to calculate conditional probability using probabilities of B given A, A, and B.

The probability that the complement of A will occur is 1 minus the probability that event A will occur.

Mutually exclusive events have zero probability of occurring together.

The word "or" is associated with the addition rule, "and" is associated with the multiplication rule.

Dependent events are when one event affects the probability of the other event occurring.

Order matters when calculating the probability of A then B vs B then A.

Pick the right probability formula based on whether events are occurring together, in sequence, are conditional, etc.

Transcripts

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