Conditional probabilitygiven that…

0:03New information, new odds

Pick a stranger at random. Are they over six feet tall? Chances are small — call it one in ten. Now one fact arrives: this person plays professional basketball. Nothing about them changed, yet the same question deserves a new answer — closer to four in five. A single piece of information just moved a probability.

0:25Shrinking the universe

Here is what that fact really does. Line up one hundred people. The phrase given that they play basketball is an instruction: keep the basketball players and set everyone else aside. Five people qualify, so the other ninety-five drop out of the problem. Your universe — the group you count in — just shrank from one hundred people down to five.

0:48Notation and the rule

Probability has a shorthand for this. P of tall, then a vertical bar, then basketball — read the bar as given. To evaluate it, count inside the shrunken universe: of the five remaining players, four are also tall. Four out of five — eighty percent. In general, the probability of tall given basketball equals tall-and-basketball, divided by basketball.

1:17Flipping the condition

Now flip it. Given that someone is over six feet, do they play basketball? The instruction changed, so the universe changes with it: shrink to the ten tall people. Inside that group, four play. Four out of ten — forty percent. Same two facts, opposite order, a different number. Basketball given tall is not tall given basketball.

1:43Hear, shrink, count

So one fact moved the odds from one in ten to four in five — not because anyone grew taller, but because the word given redrew the world before the counting started. That is conditional probability: a probability computed in the smaller world the evidence leaves behind. Hear the given. Shrink the universe to match it. Count inside what remains. New evidence, new world, new odds.