Probability Class 10 ||Maths|| Chapter 14 NCERT Notes

Probability Class 10 ||Maths|| Chapter 14 NCERT Notes

Introduction to Probability:

The chapter introduces the idea of empirical (experimental) probability, which is based on actual experiments or observations.

Definitions and Key Terms:

1. Trial:
   Any particular performance of a random experiment is called a trial. For example, tossing a coin, rolling a die, or drawing a card from a deck is a trial.

2. Random Experiment:
   An experiment in which all possible outcomes are known but the exact outcome cannot be predicted in advance. Examples include tossing a coin, rolling a dice, drawing a card from a deck, etc.

3. Outcome:
   The result of a random experiment. For example, when tossing a coin, the outcomes could be either "Heads" or "Tails."

4. Sample Space (S):
   The set of all possible outcomes of a random experiment. For example, when rolling a die, the sample space is {1, 2, 3, 4, 5, 6}.

5. Event (E):
   An event is a collection of one or more outcomes of a random experiment. It is a subset of the sample space. For example, in a dice-rolling experiment, the event of getting an even number is {2, 4, 6}.

6. Favorable Outcomes:
   These are the outcomes that correspond to a particular event of interest. For example, if the event is getting an even number on a die, then the favorable outcomes are {2, 4, 6}.

Probability Formula:

If the sample space of a random experiment has n equally likely outcomes, and m outcomes are favorable to event E, then the probability of event E is given by:

$$P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}=\frac{m}{n}$$

• P(E): Probability of event E occurring
• 0 ≤ P(E) ≤ 1: Probability ranges from 0 (impossible event) to 1 (certain event)

Types of Events:

1. Impossible Event:
   An event that cannot happen has a probability of 0. Example: Getting a 7 when rolling a standard die.

2. Sure Event:
   An event that is certain to happen has a probability of 1. Example: Getting a number less than 7 when rolling a die.

3. Complementary Events:
   For any event E, the event E’ (not E) represents all outcomes in which event E does not happen. The sum of probabilities of an event and its complement is always 1.

   $$P(E)+P(E^{\mathrm{\prime }})=1$$

   Example: If E is the event of getting an even number when rolling a die, then E' (complement) is the event of getting an odd number.

4. Equally Likely Events:
   Events that have the same probability of occurring. For example, when tossing a fair coin, getting heads and tails are equally likely events.

Important Examples:

1. Tossing a Coin:

   • Sample Space (S): {Head (H), Tail (T)}
   • Probability of getting a Head (P(H)):$$P(H)=\frac{1}{2}$$
   • Probability of getting a Tail (P(T)):$$P(T)=\frac{1}{2}$$

2. Rolling a Die:

   • Sample Space (S): {1, 2, 3, 4, 5, 6}
   • Probability of getting a number 4 (P(4)):$$P(4)=\frac{1}{6}$$
   • Probability of getting an even number:
     Favorable outcomes = {2, 4, 6}$$P(\text{even number})=\frac{3}{6}=\frac{1}{2}$$

3. Drawing a Card from a Deck: A standard deck of 52 playing cards consists of 4 suits: Hearts, Diamonds, Clubs, and Spades. Each suit has 13 cards.

   • Probability of drawing an Ace:$$P(\text{Ace})=\frac{4}{52}=\frac{1}{13}$$
   • Probability of drawing a card that is not a King:$$P(\text{Not King})=1-P(\text{King})=1-\frac{4}{52}=\frac{48}{52}=\frac{12}{13}$$

Empirical Probability (Experimental Probability):

In cases where the outcomes are not equally likely or are determined by experimentation, we use empirical probability. This is based on actual data collected from experiments.

The empirical probability of an event is calculated as:

$$P(E)=\frac{\text{Number of times event E occurred}}{\text{Total number of trials conducted}}$$

For example, if you toss a coin 100 times and get heads 55 times, the empirical probability of getting heads is:

$$P(\text{Heads})=\frac{55}{100}=0.55$$

Theoretical vs. Experimental Probability:

• Theoretical Probability: Based on the assumption that all outcomes are equally likely, calculated using the probability formula.
• Experimental Probability: Based on actual data or experiments and may differ from theoretical probability if the experiment is repeated.

Key Points to Remember:

1. The probability of an event E is always between 0 and 1:$$0\le P(E)\le 1$$
2. The sum of probabilities of all outcomes of a random experiment is 1. For example, when rolling a die:$$P(1)+P(2)+P(3)+P(4)+P(5)+P(6)=1$$
3. Complementary events are mutually exclusive and exhaustive:$$P(E)+P(E^{\mathrm{\prime }})=1$$
4. For equally likely outcomes, the probability of an event is the ratio of favorable outcomes to the total number of outcomes.

NCERT Questions and Problems:

The chapter provides multiple solved examples and exercises. Here are a few types of questions you may encounter:

1. Direct Probability Calculations:

   • Based on events like rolling a die, drawing cards, tossing coins, etc.
   • Example: What is the probability of getting a number greater than 4 when rolling a die?

2. Complementary Events:

   • Finding the probability of events by using their complementary event.
   • Example: If the probability of raining tomorrow is 0.3, what is the probability that it will not rain?

3. Empirical Probability Problems:

   • Involving actual experimental data.
   • Example: Out of 200 trials, a particular event occurred 60 times. Find the empirical probability of the event.

This wraps up the in-depth notes on Chapter 14: Probability from Class 10 NCERT Maths. If you want to explore sample problems, solved examples, or have doubts in specific sections, feel free to ask!