Conditional Probability and Independence

- Conditional Probability

In probability theory, conditional probability is the likelihood of an event occurring, given that another event has already occurred. 

Conditional probability can involve both dependent and independent events:

• Dependent events: When the first event influences the second event. For example, pulling two aces out of a deck of cards.
• Independent events: When one event occurring does not affect the probability that the other event will occur. For example, flipping a coin twice.

- Bayes' Theorem

Bayes' Theorem, named after 18th-century British mathematician Thomas Bayes, is a mathematical formula for determining conditional probability. 

In Bayesian analysis, conditional probabilities are also known as posterior probability of given.

Bayes' theorem is also known as the formula for the likelihood of “causes”. It describes the probability of an event based on the prior knowledge of the conditions that might be related to the event.

- Conditional Probability Formula

The multiplication rule of probability states that the probability of occurrence of both events A and B is equal to the product of the probability of B occurring and the conditional probability that event A occurring given that event B occurs. 

The multiplication rule can be written as P(A∩B)=P(B)⋅P(A|B).

- Examples of Conditional Probability

Here are some examples of conditional probability in real life:

• Weather forecasting: Meteorologists use conditional probability to predict the likelihood of future weather conditions based on current conditions. For example, they might calculate the probability of rain if it's cloudy outside. The probability that it will rain given that it is cloudy out, is 0.6 or 60%. This is a simplified example, but in real life weather forecasters use computer programs to take in data on current weather conditions and use conditional probability to calculate the likelihood of future weather conditions.
• Sports betting: Sports betting companies use conditional probability to set the odds that particular teams win their game.
• Selling a TV: The probability of selling a TV on a normal day might be 30%, but if the day is Diwali, the conditional probability of selling a TV might be 70%.
• Re-election of a president: The re-election of a president depends on the voting preference of voters, the success of television advertising, and the probability of the opponent making gaffes during debates.
• Team scoring: The probability of a team scoring better in the next match if they have a former Olympian for a coach is a conditional probability.

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- Conditional Independence

Conditional probability is the likelihood of an event occurring based on the occurrence of another event. Independence is when one event occurring doesn't affect the probability of another event occurring. 

In mathematical terms, independence is expressed as P(B|A)=P(B) and P(A|B)=P(A). The probability of both events A and B happening is expressed as P(A AND B) = P(A)P(B). 

For example, if a card is randomly drawn from a standard 52-card deck, the probability of the card being a queen is independent from the probability of the card being a heart.  

Conditional independence is when a condition (or outcome of one event) is unrelated to the probabilities of a different event's outcomes. 

The conditional formula for independence is P(A|B and C) = P(A|C).

Conditional independence is a special case of conditional probability. It describes situations where an observation is irrelevant or redundant when evaluating the certainty of a hypothesis. 

Conditional independence can also be proven by showing that P(A and B|C) = P(A|C)P(B|C).

[More to come ...]