Elementary Types of Events and Properties of Probability

The outcomes satisfying a particular condition are called favourable outcomes.

A set of favourable outcomes of a given sample space is an ‘event’. Event is a subset of the sample space.

• Events are denoted by capital letters A, B, C, D etc.
• The number of elements in event A is written n(A).
• Since A ⊆ S, always 0 ≤ n(A) ≤ n(S).
• Example: two coins tossed, A = getting at least one tail → A = {TT, TH, HT}, n(A) = 3.

Examples use one die: \[ S = \{1, 2, 3, 4, 5, 6\}, \; n(S) = 6. \]

\[ \begin{array}{|l|l|} \hline \textbf{TYPE} & \textbf{MEANING AND EXAMPLE} \\ \hline \text{Certain / Sure event} & \begin{array}{l} \text{Contains every outcome, } A = S. \\ \text{“Number } \leq 6\text{”} \rightarrow \{1, 2, 3, 4, 5, 6\}, \; n = 6 \end{array} \\ \hline \text{Impossible event} & \begin{array}{l} \text{Contains no outcome, } A = \{ \; \} = \phi. \\ \text{“Number 7”} \rightarrow \phi, \; n = 0 \end{array} \\ \hline \text{Simple / Elementary event} & \begin{array}{l} \text{Contains exactly one outcome.} \\ \text{“Getting 4”} \rightarrow \{4\}, \; n = 1 \end{array} \\ \hline \text{Complement of an event} & \begin{array}{l} \text{All outcomes of } S \text{ not in } A \text{; written } A'. \\ n(A') = n(S) - n(A). \\ A = \text{even} = \{2, 4, 6\} \rightarrow A' = \{1, 3, 5\} \end{array} \\ \hline \text{Mutually exclusive events} & \begin{array}{l} \text{Cannot happen together; } A \cap B = \phi. \\ A = \{1, 2\}, \; B = \{5, 6\} \end{array} \\ \hline \text{Exhaustive events} & \begin{array}{l} \text{Together cover the whole sample space;} \\ A \cup B = S. \\ A = \{1, 2, 3, 4\}, \; B = \{3, 4, 5, 6\} \end{array} \\ \hline \end{array} \]

Two coins tossed. Most marks are lost by misreading these words.

\[ \begin{array}{|l|l|} \hline \textbf{CONDITION} & \textbf{MEANING} \rightarrow \textbf{SET} \\ \hline \text{At least one tail} & \text{1 or more tails} \rightarrow \{HT, TH, TT\}, \; n = 3 \\ \hline \text{At most one tail} & \text{0 or 1 tail} \rightarrow \{HH, HT, TH\}, \; n = 3 \\ \hline \text{Only one head} & \text{Exactly 1 head} \rightarrow \{HT, TH\}, \; n = 2 \\ \hline \text{No head} & \text{0 heads} \rightarrow \{TT\}, \; n = 1 \\ \hline \end{array} \]

The sample space is the full attendance register of a class. An event is a list you make from that register by a rule: “students who wear glasses”. The list can be empty, can be the whole register, or anything in between, but it can never contain a name that isn't in the register.

Two coins are tossed simultaneously. Write the sample space ( \[ S \] ) and number of sample points \[ n(S). \] Also write the following events in the set form and write the number of sample points in each event.

Solution : If two coins are tossed simultaneously,

\[ S = \{HH, HT, TH, TT\}, \; n(S) = 4 \]

i. Condition for event \[ A \] : to get at least one tail.

\[ A = \{HT, TH, TT\} \quad n(A) = 3 \]

ii. Condition for event \[ B \] : to get only one head.

\[ B = \{HT, TH\} \quad n(B) = 2 \]

iii. Condition for event \[ C \] : to get at most one tail.

\[ C = \{HH, HT, TH\} \quad n(C) = 3 \]

iv. Condition for event \[ D \] : to get no head.

\[ D = \{TT\} \quad n(D) = 1 \]

• The sample space \[ S \] contains all possible outcomes.
• An event is a subset of the sample space.
• A simple event has one outcome; a compound event has more than one outcome.
• A certain event has probability \[ 1; \] an impossible event has probability \[ 0. \]
• The complement of \[ A \] is \[ A', \] and \[ P(A') = 1 - P(A). \]
• Mutually exclusive events cannot occur together: \[ A \cap B = \varnothing. \]
• Exhaustive events together cover the entire sample space: \[ A \cup B = S. \]
• For equally likely outcomes, \[ P(A) = \frac{n(A)}{n(S)}. \]
• For any two events, \[ P(A \cup B) = P(A) + P(B) - P(A \cap B). \]