Conditional Probability

Definition

Conditional probability measures the probability of one event given that another event is known.

Written as:

P(A | B)

Read as:

probability of A given B

Conditional probability updates the probability of one thing using known information.

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Simple Example

Let:

A = it rains
B = sky is cloudy

Then:

P(A | B)

means:

probability it rains, given that the sky is cloudy

This may be higher than the general probability of rain.

Knowing context can change probability.

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Conditional vs Unconditional Probability

Unconditional probability:

P(A)

means probability of A without extra information.

Conditional probability:

P(A | B)

means probability of A after knowing B.

Conditional probability is probability under context.

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Formula

Conditional probability can be written as:

P(A | B) = P(A and B) / P(B)

This means:

among cases where B happens,
what fraction also have A?

Conditioning restricts attention to the world where B is true.

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Example With Counts

Suppose:

100 messages total
40 are bug reports
20 are urgent
10 are both bug reports and urgent

Probability a message is urgent given it is a bug report:

P(urgent | bug) = 10 / 40 = 0.25

We divide by bug reports, not by all messages.

The condition changes the reference group.

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Why Conditional Probability Matters For Learning

Prediction often asks for conditional probability.

Example:

P(label | input)

Read as:

probability of a label given the input

This is what many classifiers try to estimate.

Prediction is often conditional probability estimation.

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Conditional Probability And Context

The same outcome can have different probabilities under different conditions.

Example:

P(rain)
P(rain | cloudy)
P(rain | clear sky)

Each answers a different question.

Probability depends on what information is being conditioned on.

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Final Summary

• Conditional probability measures probability given known information.
• It is written as P(A | B).
• It differs from unconditional probability P(A).
• Conditioning changes the reference group.
• Prediction often asks for P(label | input).
• Conditional probability connects probability to context.