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Glacius
ProbabilityConcept reference

Probability multiplication

The multiplication rule expresses a two-event intersection as a positive conditioning probability times the matching conditional probability.

2 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. An intersection requires both events on the same trial
  3. A closer look
  4. P(B | A) can differ from P(B)
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections

01An intersection requires both events on the same trial#

An intersection requires both events on the same trial. A conditional probability describes the share inside one event; multiply by that event’s mass to find the joint mass.

A contains mass 0.4. If B takes 0.75 of A, their shared mass is three quarters of 0.4: 0.3.

A has mass .4. B occupies .75 of A, so the joint is .3 and A outside B is .1. Both rows use the same probability scale; the lower full strip has length .4.A has mass .4. B occupies .75 of A, so the joint is .3 and A outside B is .1. Both rows use the same probability scale; the lower full strip has length .4.
Figure 1A has mass .4. B occupies .75 of A, so the joint is .3 and A outside B is .1. Both rows use the same probability scale; the lower full strip has length .4.
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For P(A∩B)P(A\cap B):

P(A)P(B∣A)P(A)P(B\mid A)

This form uses P(A) > 0. You can reverse the roles when P(B) > 0, using P(B) times P(A | B).

Check your reasoning

P(A)=0.5, P(B | A)=0.8. Find P(A∩B).

Show answer and explanation
0.4

0.5 × 0.8 = 0.4.

02A closer look#

Suppose a record reaches a check with chance 0.8 and passes it with chance 0.25 among records that reach it. Reaching and passing has chance 0.8 × 0.25 = 0.2.

Check your reasoning

A record passes stage 1 with chance 0.6. Among those passes, stage 2 passes with chance 0.5. Find the chance of passing both.

Show answer and explanation
0.3

Scale the within-pass share: 0.6 × 0.5 = 0.3.

03P(B | A) can differ from P(B)#

P(B | A) can differ from P(B). Multiplying two marginal probabilities needs additional information; the matching conditional factor already accounts for the restriction to A.

Check your reasoning

P(A)=0.3, P(B)=0.5, P(B | A)=0.4. A learner gives joint 0.15. Repair it.

  1. A0.15
  2. B0.4
  3. C0.12
Show answer and explanation
0.12

Use P(B | A): 0.3 × 0.4 = 0.12.

Key takeaway

Multiply the conditioning event’s probability by the share inside it. Keep the conditional direction matched to that first factor.

  • Compute a joint using its conditional factor.

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Probability multiplication.” Math behind ML. /learn/p-multiply