Probability Basics: Sample Space and Events | NIMCET

Quick answer: A sample space S is the set of all possible outcomes of a random experiment, and an event is any subset of S. If all outcomes are equally likely, the classical probability of an event E is $P(E) = \frac{n(E)}{n(S)}$ . Probabilities lie between 0 and 1, and $P(E') = 1 - P(E)$ . Odds in favour of E are $n(E) : n(E')$ .

Key terms

Term	Meaning	Example (one die)
Random experiment	Outcome cannot be predicted with certainty	Rolling a die
Sample space S	All possible outcomes	$\{1, 2, 3, 4, 5, 6\}$
Event	A subset of S	"Even number" $= \{2, 4, 6\}$
Simple event	Exactly one outcome	$\{3\}$
Sure event	The whole of S	"Number ≤ 6"
Impossible event	The empty set	"Number 7"

Types of events

Type	Condition
Mutually exclusive	$A \cap B = \phi$
Exhaustive	$A \cup B \cup \ldots = S$
Complementary	$A' = S - A$
Equally likely	Each has the same chance

Events are sets, so the set operations apply directly: "A or B" is $A \cup B$ , "A and B" is $A \cap B$ .

Classical definition

$P(E) = \frac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}} = \frac{n(E)}{n(S)}$

Property	Result
Range	$0 \le P(E) \le 1$
Sure event	$P(S) = 1$
Impossible event	$P(\phi) = 0$
Complement	$P(E') = 1 - P(E)$
Odds in favour	$P(E) : P(E')$
From odds a : b in favour	$P(E) = \frac{a}{a + b}$

Standard sample spaces

Experiment	$n(S)$
n coins tossed	$2^n$
n dice rolled	$6^n$
One card from a pack	52
r cards from a pack	$^{52}C_r$

A pack of 52 cards: 4 suits of 13 cards; 26 red and 26 black; 12 face cards (J, Q, K); 4 aces.

Sums with two dice:

Sum	2	3	4	5	6	7	8	9	10	11	12
Ways	1	2	3	4	5	6	5	4	3	2	1

Worked examples

Example 1: coins

Three coins are tossed. Find the probability of at least two heads.

Favourable: HHH, HHT, HTH, THH, so $P = \frac{4}{8} = \frac{1}{2}$ .

Example 2: two dice

Find the probability that the sum on two dice is 8.

From the table, 5 of 36 outcomes: $P = \frac{5}{36}$ .

Example 3: cards

Two cards are drawn from a pack. Find the probability that both are kings.

$P = \frac{^4C_2}{^{52}C_2} = \frac{6}{1326} = \frac{1}{221}$

Example 4: complement

Find the probability of at least one six in 3 rolls of a die.

$1 - \left(\frac{5}{6}\right)^3 = 1 - \frac{125}{216} = \frac{91}{216}$

Example 5: odds

The odds in favour of an event are 3 : 5. Find its probability.

$P = \frac{3}{8}$ .

Summary table

Idea	Key result
Classical probability	$\frac{n(E)}{n(S)}$
Complement	$1 - P(E)$
Odds a : b in favour	$\frac{a}{a + b}$
n coins	$2^n$ outcomes
Two dice, sum 7	$\frac{6}{36} = \frac{1}{6}$

Exam tips

"At least one" problems are almost always fastest with the complement.
Use combinations ( $^nC_r$ ) for drawing several objects at once; see combinations.
Memorise the two-dice sum table.

Next, read the addition theorem of probability, or see the NIMCET Probability and Statistics chapter.

Common mistakes

Using the classical formula when outcomes are not equally likely.
Counting HT and TH as one outcome.
Confusing odds with probability.
Forgetting that a pack has 12 face cards, not 16 (aces are not face cards).

Frequently Asked Questions

What is the classical definition of probability?

P(E) = n(E)/n(S), the number of favourable outcomes divided by the total number of equally likely outcomes.

What is a sample space?

The set of all possible outcomes of a random experiment.

What are mutually exclusive events?

Events that cannot happen together, so A ∩ B = φ.

How do you convert odds to probability?

If the odds in favour are a : b, the probability is a/(a + b).

What is the probability of getting a sum of 7 with two dice?

It is 6/36 = 1/6.