Theoretical Probability, Elementary Events, and Complementary Events for CBSE Class 10

Every day, we make statements fraught with uncertainty: "It will probably rain this evening," "There is a fifty-fifty chance India will win the cricket toss," or "It is impossible for a standard die to roll an eight." But can human uncertainty be measured with mathematical exactness?

In the 17th century, mathematicians Pierre de Fermat, Blaise Pascal, and later Pierre-Simon Laplace transformed uncertainty into a rigorous science: Probability Theory. In Class 9, you approached probability experimentally by recording empirical frequencies from trials. In CBSE Class 10 Mathematics, Chapter 14 (Probability) introduces the powerful Theoretical (Classical) Approach, where probabilities are calculated before performing an experiment based entirely on the geometry and symmetry of sample spaces.

What You Will Learn

• Experimental (Empirical) Probability vs. Theoretical (Classical) Probability
• The concept of Equally Likely Outcomes
• Pierre-Simon Laplace's Classical Definition of Probability: P(E) = rac{n(E)}{n(S)}
• What is an Elementary Event?
• The fundamental axiom: The sum of probabilities of all elementary events equals $1$
• Complementary Events: The "Not $E$" principle ($P(E) + P(\bar{E}) = 1$)
• Impossible events ($P = 0$) and Sure / Certain events ($P = 1$)
• The strict bounds of probability: $0 \le P(E) \le 1$
• Solved CBSE board examination problems and common traps

1. Theoretical vs. Experimental Probability

2. Equally Likely Outcomes and Classical Definition

When you roll an unbiased six-sided die, you have no physical reason to expect a $6$ to appear more often than a $1, 2, 3, 4,$ or $5$. All six faces are symmetrically identical. Outcomes are said to be equally likely if each outcome has the exact same chance of occurring as any other.

Laplace's Classical Definition of Probability

If an experiment has $n(S)$ equally likely, mutually exclusive outcomes, and $n(E)$ of these outcomes are favourable to the occurrence of an event $E$, then the theoretical probability of event $E$, denoted by $P(E)$, is defined as: $\mathbf{P(E) = \frac{\text{Number of outcomes favourable to } E}{\text{Number of all possible outcomes of the experiment}} = \frac{n(E)}{n(S)}}$

Where:

• Sample Space ($S$): The set of all possible outcomes of a random experiment.
• $n(S)$: Total number of possible outcomes.
• $n(E)$: Number of outcomes favourable to the event $E$.

3. Elementary Events and the Sum-to-One Rule

Definition

An event having only one single outcome of the experiment is called an elementary event.

Example: In tossing a single coin, the event of getting a Head ($E_1 = \{H\}$) is an elementary event with $P(E_1) = 1/2$. The event of getting a Tail ($E_2 = \{T\}$) is an elementary event with $P(E_2) = 1/2$.

The Fundamental Probability Axiom:

The sum of the probabilities of all the elementary events of an experiment is ALWAYS equal to $1$! $\mathbf{\sum P(E_i) = P(E_1) + P(E_2) + \dots + P(E_n) = 1}$

Verification on a 6-sided die: $P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{6}{6} = 1$

4. Complementary Events: The "Not E" Rule

For any event $E$, there exists a corresponding complementary event, denoted by $\bar{E}$ (or $E'$), representing the occurrence of "not $E$".

                           Total Sample Space S (Probability = 1)
                  +---------------------------------------------------+
                  |                 +---------------+                 |
                  |                 |    Event E    |                 |
                  |                 |     P(E)      |                 |
                  |                 +---------------+                 |
                  |             Event "Not E" (E)                     |
                  |             P(E) = 1 - P(E)                       |
                  +---------------------------------------------------+

The Complementary Formula:

$\mathbf{P(E) + P(\bar{E}) = 1}$ $\mathbf{P(\bar{E}) = 1 - P(E)}$

Example: If the probability of winning a game is $0.62$, the probability of losing (not winning) is: $P(\text{Losing}) = 1 - 0.62 = \mathbf{0.38}$

5. Impossible Events, Sure Events, and Probability Bounds

1. Impossible Event: An event that has zero chance of occurring.
   • Example: Rolling a number $7$ or $8$ on a standard single six-sided die ($n(E) = 0$).
   • The probability of an impossible event is always $0$: $\mathbf{P(\text{Impossible Event}) = 0}$
2. Sure Event (Certain Event): An event that is guaranteed to happen in every trial.
   • Example: Rolling a number less than $7$ on a standard die (all outcomes $1, 2, 3, 4, 5, 6$ are favourable).
   • The probability of a sure event is always $1$: $\mathbf{P(\text{Sure Event}) = 1}$

The Universal Range of Probability:

<u>The probability of any event $E$ is always a number between $0$ and $1$ (inclusive):</u> $\mathbf{0 \le P(E) \le 1}$

• Probability can NEVER be negative (e.g., $-1.5$ is impossible).
• Probability can NEVER be greater than 1 (e.g., $1.4$ or $150\%$ is impossible).

6. Solved CBSE Board Examination Problems

Solved Example 1: Identifying Valid Probabilities (CBSE 1-Mark MCQ)

Problem: Which of the following cannot be the probability of an event? (A) $2/3$
(B) $-1.5$
(C) $15\%$
(D) $0.7$

Solution:

1. A valid probability must satisfy $0 \le P(E) \le 1$.
2. Check each option:
   • (A) $2/3 \approx 0.67$ (Valid: lies between $0$ and $1$).
   • (B) $-1.5$ is negative! Probability can never be negative.
   • (C) $15\% = \frac{15}{100} = 0.15$ (Valid).
   • (D) $0.7$ (Valid).
3. Therefore, <u>the correct option is (B) $-1.5$</u>.

Solved Example 2: Complementary Event Calculation

Problem: If $P(E) = 0.05$, what is the probability of 'not $E$'?

Solution:

1. Using the complementary event formula: $P(\text{not } E) = 1 - P(E)$
2. Substitute $P(E) = 0.05$: $P(\text{not } E) = 1 - 0.05 = \mathbf{0.95}$
3. Therefore, <u>the probability of 'not $E$' is $0.95$</u>.

7. Summary and Examination Tips

Exam Tip: Whenever an event has many complex favourable cases, check if the complementary event ("not $E$") is much simpler to calculate! Use $P(E) = 1 - P(\bar{E})$ to save valuable exam time.

Common Mistake: Expressing probability as a ratio greater than 1, such as writing $\frac{6}{4}$. The numerator ($n(E)$) can NEVER be larger than the denominator ($n(S)$)!