Probability — Part 2: The axiomatic approach: visual lesson, CBSE Class 11 Mathematics

Lesson

From counting to rules

Counting favourable outcomes is not always possible. The axiomatic approach gives a few rules that every probability must obey, and everything else follows from them.

In this lesson you will

State the three axioms and use them to assign and test probabilities
Find probabilities of events with equally likely and with weighted outcomes
Use P(not A) = 1 − P(A) and the addition theorem for P(A or B)
Solve problems on neither, A but not B and exactly one of two events

Pick counting or weights first, then the rule for the event you need.

Concept

The three axioms

A probability $P$ assigns to each event a number so that $P(E) \ge 0$ , $P(S) = 1$ , and $P(A \cup B) = P(A) + P(B)$ whenever $A \cap B = \varnothing$ .

P(E) \ge 0,\quad P(S) = 1

P(A \cup B) = P(A) + P(B),\ A \cap B = \varnothing

Everything else in the chapter follows from these three statements.
Axiom	In symbols	In words
Non-negative	$P(E) \ge 0$	No event has a negative probability
Total one	$P(S) = 1$	Something in the sample space is certain to happen
Additive	$P(A \cup B) = P(A) + P(B)\ \text{if } A \cap B = \varnothing$	Probabilities of mutually exclusive events add

Concept

Assigning probabilities to outcomes

List the elementary events and give each a probability, so that every probability is non-negative and the total is $1$ . The probability of an event is the sum over its outcomes.

The elementary probabilities must add up to $1$.
Outcome	$a$	$b$	$c$	$d$	Total
Probability	0.15	0.25	$x$	$2x$	$0.4 + 3x = 1$
With $x = 0.2$	0.15	0.25	0.2	0.4	1

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