Class 10 NCERT Maths Chapter 14: Probability
Probability is the measure of how likely an event is to occur. In this chapter, we focus on empirical probability (based on experimental data) and theoretical probability (based on reasoning).
1. Basic Terms
- Trial – An action with an uncertain result (e.g., tossing a coin).
- Outcome – A possible result of a trial (e.g., Head or Tail).
- Event – A collection of outcomes (e.g., getting a Head).
- Sample Space – The set of all possible outcomes, denoted by \(S\).
2. Theoretical Probability
If all outcomes are equally likely, the probability of an event \(E\) is:
\[ P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}} \]
Example: A card is drawn from a deck of 52 cards. Probability of getting a king: \[ P(E) = \frac{4}{52} = \frac{1}{13} \]
3. Complementary Events
For any event \(E\): \[ P(E) + P(\overline{E}) = 1 \] where \(\overline{E}\) = event “not E”.
Example: Probability of getting an even number on a die: \[ P(E) = \frac{3}{6} = \frac{1}{2} \] Probability of getting an odd number: \[ P(\overline{E}) = 1 - \frac{1}{2} = \frac{1}{2} \]
4. Some Common Cases
| Experiment | Sample Space \(S\) | Example Event | Probability |
|---|---|---|---|
| Coin Toss | \(\{H, T\}\) | Getting Head | \(\frac{1}{2}\) |
| Rolling a Die | \(\{1, 2, 3, 4, 5, 6\}\) | Getting a number > 4 | \(\frac{2}{6} = \frac{1}{3}\) |
| Deck of Cards | 52 cards | Drawing a red card | \(\frac{26}{52} = \frac{1}{2}\) |
5. Empirical Probability
When we perform an experiment repeatedly, the probability of an event \(E\) can be estimated as: \[ P(E) \approx \frac{\text{Number of times E occurs}}{\text{Total number of trials}} \] This is called empirical probability.
6. Important Points
- Probability values are always between 0 and 1.
- \(P(E) = 0\) means the event is impossible.
- \(P(E) = 1\) means the event is certain.
- More trials → Empirical probability gets closer to theoretical probability.