Multiplication Theorem of Probability: Formula, Rules and Examples
Multiplication Theorem of Probability gives the probability of the simultaneous occurrence of two or more events. It expresses a joint probability using the probability of one event and the conditional probability of the next event. This page covers its formulas, independent and dependent events, sequential applications, solved examples and common probability problems.
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What is the Multiplication Theorem of Probability?
Here, P(B|A) is the probability of B when A has already occurred. The equivalent form is P(A ∩ B) = P(B)P(A|B). For several events, P(A₁ ∩ A₂ ∩ A₃) = P(A₁)P(A₂|A₁)P(A₃|A₁ ∩ A₂).
Multiplication Theorem of Probability Video Lessons
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Independent Events and Multiplication Theorem
Learn how to identify independent events and apply the multiplication theorem to calculate the probability of their joint occurrence.
Multiplication Theorem of Probability Formula & Tricks
Important Formulas
The joint probability of A and B equals the probability of A multiplied by the conditional probability of B after A occurs.
The same joint probability can be calculated by taking B first and then finding the conditional probability of A given B.
If A and B are independent, the occurrence of one does not change the probability of the other.
For successive events, each factor is the probability of the next event given that all preceding events have occurred.
Quick Tricks
Since P(A ∩ B) can be written in either order, choose P(A)P(B|A) or P(B)P(A|B) according to the easier conditional probability.
When an object is selected and not replaced, the total number of remaining objects decreases before calculating the next conditional probability.
Use P(A)P(B) only for independent events. If the first event changes the second probability, use P(A)P(B|A).
Multiplication Theorem of Probability Concepts
Conditional probability form
The intersection A ∩ B means that both A and B occur. The conditional factor P(B|A) is calculated after assuming that A has occurred. If P(A) is non-zero, then P(B|A) = P(A ∩ B)/P(A).
Successive dependent events
For three events, P(A ∩ B ∩ C) = P(A)P(B|A)P(C|A ∩ B). In sampling without replacement, the numerator and denominator change after every selection.
Independent events
Events are independent when the occurrence of one event does not alter the probability of the other. Repeated tosses of a fair coin are independent, provided the tosses do not influence one another.
Mutually exclusive versus independent events
If A and B are mutually exclusive, then P(A ∩ B) = 0. If both events have positive probability, mutually exclusive events cannot be independent because P(A)P(B) is then positive.
Tree-based multiplication of probabilities
For alternative paths, calculate each path probability separately and add the required path probabilities. Every branch probability must be conditional on the outcomes already shown on that path.
Multiplication Theorem of Probability Data Table
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Comparison of event relationships
The multiplication rule changes according to whether the events affect one another.
| Event relationship | Condition | Joint probability |
|---|---|---|
| General events | B may depend on A | P(A ∩ B) = P(A)P(B|A) |
| Independent events | P(B|A) = P(B) | P(A ∩ B) = P(A)P(B) |
| Mutually exclusive events | A and B cannot occur together | P(A ∩ B) = 0 |
Note: Events with positive probabilities cannot be both mutually exclusive and independent.
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Multiplication Theorem of Probability: Revision Points
Use these rules to revise joint probabilities and sequential event calculations.
- P(A ∩ B) = P(A)P(B|A) = P(B)P(A|B).
- For independent events, P(A ∩ B) = P(A)P(B).
- For three events, multiply the first probability by each successive conditional probability.
- Without replacement, update the total and favourable outcomes after every selection.
- Mutually exclusive events have joint probability zero.
- In a probability tree, multiply along a path and add probabilities of alternative required paths.
Multiplication Theorem of Probability FAQs
What is the multiplication theorem of probability formula?
The formula is P(A ∩ B) = P(A)P(B|A) = P(B)P(A|B), where P(B|A) is the probability of B given A.
How is the formula used for independent events?
For independent events, P(B|A) = P(B). Therefore, P(A ∩ B) = P(A)P(B).
What is the probability of drawing two aces consecutively without replacement?
It is 4/52 × 3/51 = 1/221, because there are 4 aces initially and 3 aces after one ace is drawn.
How do you calculate the probability of three successive events?
Use P(A ∩ B ∩ C) = P(A)P(B|A)P(C|A ∩ B). Each later factor includes the outcomes of the earlier events.
Can mutually exclusive events be independent?
Only in the special case where at least one event has probability zero. If both probabilities are positive, mutually exclusive events are not independent.
What is the difference between with-replacement and without-replacement probability?
With replacement, the composition and probabilities remain unchanged. Without replacement, the total and favourable counts change after each selection.