Elements of probability and Tree diagram — lesson. Mathematics CBSE, Class 9 New.

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Sample space and events

The sample space is the set of all possible outcomes of a random experiment. It is represented by $S$. Each individual outcome is called an element of the sample space.

Rules for a valid sample space:

The sample space must contain all possible outcomes.
No outcome should be repeated.

The number of elements in the sample space is denoted by $n(S)$.

Experiment	Sample space	$n(S)$
Tossing $1$ coin	$\{H, T\}$	$2$
Tossing $2$ coins	$\{HH, HT, TH, TT\}$	$4$
Rolling a die	$\{1, 2, 3, 4, 5, 6\}$	$6$
Rolling $2$ dice	$\{\begin{array}{l} (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) \\ (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) \\ (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6) \\ (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) \\ (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) \\ (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \end{array}\}$	$36$

Event:

An event is a collection of one or more outcomes from the sample space. Thus, every event is a subset of the sample space.

Example:

Getting an even number from rolling a die.

Solution:

Sample space, $S = \{1, 2, 3, 4, 5, 6\}$

Getting an even number, $E = \{2, 4, 6\}$

Here,

E \subseteq S

Tree diagram

A tree diagram is a pictorial method for showing all possible outcomes of a multi-step experiment.

It helps us:

organise outcomes systematically,
avoid missing any outcome,
calculate probabilities easily.

Let's toss a coin and write the possible outcomes(Head or Tail) using a tree diagram.

So, the sample space of tossing a coin is $S = \{H,T\}$.

Let's toss $2$ coins and write all the possible outcomes of this experiment using a tree diagram.

Similarly, when we throw a die, there are $6$ possible outcomes. We can make a tree diagram for this as well.

Constructing tree diagrams:

Step 1: Draw a starting point.

Step 2: Draw a branch to list all the outcomes after the first toss/roll.

Step 3: Draw a second branch listing all possible outcomes from Step 2.

Step 4: Continue for additional steps if needed.

Step 5: Each complete path from start to the end of the last branch represents one outcome in the sample space.

Example:

Draw a tree diagram of rolling a die twice.

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Experiment	Sample space	\(n(S)\)
Tossing \(1\) coin	\(\{H, T\}\)	\(2\)
Tossing \(2\) coins	\(\{HH, HT, TH, TT\}\)	\(4\)
Rolling a die	\(\{1, 2, 3, 4, 5, 6\}\)	\(6\)
Rolling \(2\) dice	$\{\begin{array}{l} (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) \\ (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) \\ (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6) \\ (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) \\ (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) \\ (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \end{array}\}$	\(36\)

1. Elements of probability and Tree diagram

Theory: