Read it again any time, including while you work through the exercises.
A piece of information can change a probability. What is the probability that a person likes tea? And what if I first tell you this person drinks it every morning? Your estimate moves: you just did conditional probability without knowing it.
The probability of A given B, written P of A given B, is the probability that A happens when we KNOW that B has happened. The key idea: the information "B happened" shrinks the universe. We no longer reason over all possible cases, only over the cases where B occurs. The formula says exactly that: P(A given B) = P(A and B) divided by P(B).
In a two-way table of counts, it is very concrete: to compute P(A given B), you place yourself in B's row (or column), and you divide the count of "A and B" by that row's total. The denominator is no longer the grand total: that is the shrinking of the universe.
On a weighted tree, conditional probabilities live on the second-level branches. The path rule becomes: P(A and B) = P(A) × P(B given A). And by adding all the paths leading to the same event, you get its total probability.
Beware of the most widespread confusion: P(A given B) and P(B given A) are in general DIFFERENT. The probability of being a cat given that you are black is not the probability of being black given that you are a cat. This lesson trains you to spot which event is the condition, and never to flip them again.
Out of 100 people surveyed, 40 do sport, and among these 40 athletes, 30 listen to music while training. We pick a person at random. What is the probability that they listen to music given that they do sport?
- 1
univers restreint : 40 sportifs
The information "does sport" shrinks the universe: I now only look at the 40 athletes, the other 60 no longer count.
- 2
cas favorables : 30
Among these 40 people, 30 listen to music.
- 3
30/40 = 0,75
I divide the favourable count by the restricted group's total: 30 out of 40, that is 0.75.
Result: P(music given sport) = 0.75.