Wolfram Alpha:
Search by keyword:
Astronomy
Chemistry
Classical Physics
Climate Change
Cosmology
Finance and Accounting
Game Theory
General Relativity
Lagrangian and Hamiltonian Mechanics
Macroeconomics
Mathematics
Microeconomics
Particle Physics
Probability and Statistics
Programming and Computer Science
Quantum Computing
Quantum Field Theory
Quantum Mechanics
Semiconductor Reliability
Solid State Electronics
Special Relativity
Statistical Mechanics
String Theory
Superconductivity
Supersymmetry (SUSY) and Grand Unified Theory (GUT)
The Standard Model
Topology
Units, Constants and Useful Formulas
Probability Rules
-----------------
Mutually Exclusive Events
-------------------------
A and B cannot happen at the same time. Therefore:
P(A and B) ≡ P(A ∩ B) = 0
Example:
A simple coin toss.
Independent Events
------------------
A and B are not related to each other. Therefore:
P(A ∩ B) = P(A).P(B)
Clearly A and B cannot be both mutually exclusive and independent.
Example:
Probability of rolling a 6 and tossing a head is (1/6)(1/2) = 1/2
Dependent Events
----------------
When A and B, are dependent, the probability of both occurring is:
P(A ∩ B) = P(A)·P(B|A)
Example:
A card is chosen at random from a deck of cards. Without replacing
it, a second card is chosen. What is the probability that the first
card chosen is a queen and the second card chosen is a jack?
P(Q) = 4/52
P(J|Q) = 4/51
P(Q ∩ J) = (4/52)(4/51) = 4/663
Dependent events are discussed in more detail in the note on
conditional probability.
Addition Rules
--------------
If A and B ARE mutually exclusive:
P(A or B) ≡ P(A ∪ B) = P(A) + P(B)
Example:
A single die is rolled. What is the probability of rolling a
2 or a 5?
P(2 ∪ 5) = P(2) + P(5)
= 1/6 + 1/6
= 1/3
If A and B are NOT mutually exclusive:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
= P(A) + P(B) - P(A).P(B)
= P(A) + P(B)(1 - P(A))
Example:
What is the probability pulling a club or a king from a deck
of cards?
P(C ∪ K) = P(C) + P(K) - P(C ∩ K)
= 4/52 + 13/52 - 1/52 or 4/52 + (13/52)(1 - 4/52 )
= 4/13