Probability

Independence & Conditional Probability

Determining independence and computing conditional probabilities.

Intermediate20 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Probability Rules

When Knowing One Thing Changes the Probability of Another

A survey of 50 students records handedness and whether they play a musical instrument: of 45 right-handed students, 18 play an instrument; of 5 left-handed students, 3 play an instrument. Before reading on: does knowing a student is left-handed seem to change how likely they are to play an instrument, compared to the group as a whole?

Definition — Conditional Probability and Independence

P(A|B) = P(A and B)/P(B) restricts attention to only the cases where B happened, then asks what fraction of those also satisfy A. Two events are independent exactly when P(A|B) = P(A) — knowing B happened doesn't change A's probability at all.

P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}

	Plays Instrument	Doesn't Play	Total
Right-handed	18	27	45
Left-handed	3	2	5
Total	21	29	50

50 students by handedness and instrument-playing.

Worked Example — Computing and Comparing Conditional Probabilities

P(plays instrument | right-handed) = 18/45 = 0.40. P(plays instrument | left-handed) = 3/5 = 0.60. The overall (unconditional) proportion who play is 21/50 = 0.42. Since the conditional proportions (40% and 60%) differ from each other and from the overall 42%, handedness and instrument-playing don't look independent in this sample. That said, only 5 students in this sample are left-handed — a very small subgroup — so this particular comparison is far less reliable than it might look; a genuinely useful conclusion here would need a larger sample of left-handed students.

Tip

A conditional probability computed from a very small subgroup can look dramatically different from the overall rate purely by chance — always note the group size before drawing a strong conclusion from a conditional probability.

Common Mistakes

Computing P(right-handed | plays instrument) when the question actually asked for P(plays instrument | right-handed).
These are generally different quantities, computed in opposite directions — 18/45 (given right-handed, plays) is not the same as 18/21 (given plays, is right-handed) — always identify which event is being conditioned on which.

Key Takeaways

P(A|B) restricts attention to the outcomes where B occurred, then finds what fraction also satisfy A.
Independence means P(A|B) = P(A) — learning B occurred doesn't shift A's probability.
A conditional probability from a small subgroup deserves extra caution before drawing a firm conclusion from it.

Summary

This closes the unit's probability-rules topic. The next lesson builds a full probability distribution for a numerical outcome that varies by chance.

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