Probability Class 10 Mathematics Concept notes and Formula Sheet

 

Concept Notes

1. Introduction to Probability

• Probability is a measure of the likelihood of an event happening. It is expressed as a number between 0 and 1, where:
  • 0 indicates an impossible event.
  • 1 indicates a certain event.
  • The probability of any event E is denoted as P(E).

2. Random Experiment

• A random experiment is an experiment or a process for which the outcome cannot be predicted with certainty.
• Example: Tossing a coin, rolling a die, drawing a card from a deck.

3. Sample Space

• The set of all possible outcomes of a random experiment is called the sample space, denoted by S.
• Example:
  • Tossing a coin: S = {Head, Tail}
  • Rolling a die: S = {1, 2, 3, 4, 5, 6}

4. Event

• An event is a subset of the sample space. It represents one or more outcomes.
• Example: In rolling a die, the event of getting an even number is E = {2, 4, 6}.

5. Types of Events

• Sure Event: An event that is certain to occur. P(S) = 1.
• Impossible Event: An event that cannot occur. P(∅) = 0.
• Complementary Events: If E is an event, then the complement of E (denoted by E') is the event that E does not happen. P(E) + P(E') = 1.
• Mutually Exclusive Events: Two events are mutually exclusive if they cannot occur at the same time. P(A ∩ B) = 0.
• Exhaustive Events: Events are exhaustive if they cover the entire sample space.

6. Classical Definition of Probability

• If a random experiment results in n equally likely outcomes, and m of them are favorable to event E, then the probability of event E is given by: $$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{m}{n}$$

7. Empirical Probability

• Empirical probability (or experimental probability) is based on observations or actual experiments. It is calculated as: $$P(E) = \frac{\text{Number of times event } E \text{ occurs}}{\text{Total number of trials}}$$

8. Theoretical Probability

• Theoretical probability is based on reasoning or calculation rather than actual trials. It is often used when the outcomes are equally likely.

9. Important Properties of Probability

• The probability of any event lies between 0 and 1, inclusive: $0 \leq P(E) \leq 1$.
• The sum of probabilities of all mutually exclusive and exhaustive events is 1.

Formula Sheet

1. Probability of an Event:

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

2. Complementary Event:

   $$P(E') = 1 - P(E)$$

   where $P(E')$ is the probability of the event not occurring.

3. Empirical Probability:

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

4. Mutually Exclusive Events:

   $$P(A \cup B) = P(A) + P(B)$$

   if $A$ and $B$ are mutually exclusive events.

5. Addition Theorem of Probability:

   $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

   This is used when events are not mutually exclusive.

6. Total Probability:

   $$P(S) = 1$$

   where $S$ is the sample space.

Examples for Practice

1. What is the probability of getting a head when tossing a coin?
2. A die is rolled. Find the probability of getting a number greater than 4.
3. A box contains 5 red, 7 blue, and 3 green balls. If a ball is drawn at random, what is the probability that it is blue?