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Conditional probability and independence

Update probabilities on what is known. Start with what a conditional probability means and how to read it, then compute it by counting and from the formula, and build probabilities in stages with the multiplication and chain rules and probability trees. Then decide when events are independent and compute with several independent events, split a probability into cases with the law of total probability and choose what to condition on, and finish with independence once a third event is known.

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The conditional probability formula

Learn to find a conditional probability when outcomes are not equally likely, by restricting to the known event and rescaling by its probability, and why every rule of probability still holds given that event.

P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}

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Lesson 03 · The conditional probability formula

0,0

Ω

A

B

0.2

0.3

0.4

A \cap B

0.1

Step 1: The four regions

Each region of $Ω$ is labelled with its probability: $A$ only $0.2$ , both $0.1$ , $B$ only $0.3$ , and neither $0.4$ . Together they total $1$ .

Step 2: Restrict to $B$

We learn that $B$ happens, so every outcome outside $B$ is ruled out. Only the two regions of $B$ remain, and their probabilities total

P (B) = 0.1 + 0.3 = 0.4

Step 3: $B$ becomes the whole

Once we know that $B$ happens, $B$ is everything that can happen: its $0.4$ now plays the part that $1$ played before. So we rescale, measuring each region as a share of $B$ : both is $0.1$ out of $0.4$ , a quarter of $B$ , and $B$ only is $0.3$ out of $0.4$ , three quarters.

\frac{0.1}{0.4} = 0.25 \frac{0.3}{0.4} = 0.75

The shares total $1$ : a conditional probability is still a probability.

Step 4: Read off $A$

Now $A$ can happen only if $B$ happens too, so all that is left of $A$ is the overlap $A \cap B$ . $P (A ∣ B)$ is the overlap’s share of $B$ :

P (A ∣ B) = \frac{0.1}{0.4} = 0.25

This is the counting rule from before, the outcomes in both divided by the outcomes in $B$ , with probabilities in place of counts.

If $A$ and $B$ had no overlap, nothing of $A$ would be left once we restrict to $B$ , and $P (A ∣ B)$ would be $0$ .

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11 of 15 lessons

11 lessons you can take today, and 4 more being written. Expand any one to see what it covers. Each unlocks when its own prerequisites are passed.

What is conditional probability?

8 questions

Learn what a conditional probability is, how knowing that one event happens can change the chance of another, and how to write it in notation.

Covers

What conditional probability is

Reading $P(A \mid B)$

Conditional probability by counting

13 questions

Learn to find a conditional probability by counting equally likely outcomes, and why "at least one" and "a named one" give different answers.

Covers

Restricting to the known event

From a two-way table

At least one versus a named one

The conditional probability formula

Preview

9 questions

Covers

The formula from probabilities

Probability rules given an event

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The multiplication rule for probabilities

9 questions

Learn how to find the probability that two events both happen as the probability of one times the conditional probability of the other given it, and how to pair a given conditional probability with the probability of its known event.

Covers

The probability of one event followed by another

Pairing a conditional probability with its known event

The chain rule for probabilities

6 questions

Learn how to find the probability that several events all happen as a product: the probability of the first, times the probability of each later event given all the events before it.

Covers

The chain rule for probabilities

Independence of events

8 questions

Learn what it means for two events to be independent, how to judge from the setting whether they are, and why mutually exclusive events are always dependent.

Covers

What independence means

Mutually exclusive events are dependent

Independence by calculation

8 questions

Learn how to check whether two events are independent by calculating and comparing probabilities, and how to use the product rule to find the probability that two independent events both happen.

Covers

Checking independence

The product rule for independent events

Independence of several events

5 questions

Learn how to decide whether three events are independent, and why every pair being independent is not enough.

Covers

Independence of three events

Computing with several independent events

8 questions

Learn how to find the probability that some independent events happen and the others do not, and the probability that at least one of them happens.

Covers

Complements of independent events

At least one of several independent events

Conditional independence

9 questions

Learn what it means for two events to be conditionally independent given a third, and how to check it by comparing probabilities once that event is known.

Covers

What conditional independence means

Checking conditional independence

Computing with conditional independence

8 questions

Learn how to find the probability of a sequence of events when each depends only on the one before it, and how to find the probability that two conditionally independent events both happen.

Covers

Probabilities of sequences of events

The product rule given $C$

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Probability trees

In drafting

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The law of total probability

In drafting

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Choosing what to condition on

In drafting

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Independence versus conditional independence

In drafting

Where this unit sits in the curriculum

This unit is part of our Essential Probability & Statistics for ML learning path, which contains 74 lessons. Every one of them is drawn below.

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11 lessons in this unit

33 prerequisite lessons across the unit, counting every step back to the start

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More in Probability and Statistics: Sample spaces and events, Counting, Probability fundamentals

See it taught

The conditional probability formula

Step 1: The four regions

Step 2: Restrict to B

Step 3: B becomes the whole

Step 4: Read off A

11 of 15 lessons

Where this unit sits in the curriculum

Start Conditional probability and independence

Step 2: Restrict to $B$

Step 3: $B$ becomes the whole

Step 4: Read off $A$