Conditional probability

Your friend rolls two dice behind a book, peeks, and says: “I can tell you there’s at least one six.” What’s the chance it’s a double six?

Before she spoke, double six had a $\frac{1}{36}$ chance. Now it must be higher: you’ve learned something, and the boring outcomes with no six are gone. But how much higher? Many people say $\frac{1}{6}$ (“one die is a six, the other has a 1 in 6 chance”). The real answer is smaller. You’ll find it on the grid in a moment.

This is conditional probability: updating chances when you learn something. It’s how a spam filter reacts to the word “lottery”, how your phone suggests the next word, and how doctors read test results (that one is lesson 5).

The real words

• Conditional probability P(A | B) is the probability of $A$ given that $B$ happened. The vertical bar is read “given”.

  $$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$$

  (This needs $P(B) > 0$: you can’t condition on something impossible.)

  • Why: $B$ is the new world, so only the part of $A$ inside $B$ counts ($A \cap B$), and we divide by the size of the new world, $P(B)$, so that the new world adds up to 1 again.
  • With equally likely outcomes this is just counting: $\frac{|A \cap B|}{|B|} = \frac{1}{11}$ for the friend’s hint.

• Multiplication rule: multiply both sides by $P(B)$:

  $$P(A \cap B) = P(B) \cdot P(A \mid B)$$

  • Example: draw 2 cards without putting the first back. $P(\text{both aces}) = \frac{4}{52} \cdot \frac{3}{51} = \frac{1}{221}$: the second fraction is “ace given the first was an ace”, with 3 aces left in 51 cards.

  • It chains, one event at a time:

    $$\begin{aligned} &P(A \cap B \cap C) \\ &= P(A) \cdot P(B \mid A) \cdot P(C \mid A \cap B) \end{aligned}$$

    Drawing pictures of these chains gives a probability tree (lesson 6).

• Don’t flip the bar: $P(A \mid B) \ne P(B \mid A)$ in general. Almost every professional basketball player is tall, but very few tall people are professional basketball players. Mixing these two up is so common it has a name in courtrooms, the prosecutor’s fallacy, and lesson 5 shows how to get from one to the other.