Probability

Probability is a branch of mathematics that deals with the likelihood or chance of different outcomes occurring in an experiment. It is widely used in statistics, science, engineering, and everyday life to predict uncertain events.

1. Random Experiments and Events

• Random Experiment: An experiment whose outcome cannot be predicted with certainty (e.g., tossing a coin, rolling a die).
• Sample Space (S): The set of all possible outcomes of a random experiment.
• Event: A subset of the sample space. An event may contain one or more outcomes.

2. Types of Events

• Simple Event: Contains only one outcome.
• Compound Event: Contains more than one outcome.
• Impossible Event: An event that cannot occur (probability = 0).
• Sure Event: An event that is certain to occur (probability = 1).
• Mutually Exclusive Events: Events that cannot happen at the same time.
• Exhaustive Events: All possible outcomes together form exhaustive events.

3. Classical Definition of Probability

If an experiment has n equally likely, mutually exclusive, and exhaustive outcomes, and an event E can occur in m ways, then:

P(E) = Number of favourable outcomes / Total number of possible outcomes = m / n

4. Properties of Probability

• Probability of any event E: 0 ≤ P(E) ≤ 1
• Probability of impossible event: P(∅) = 0
• Probability of sure event: P(S) = 1
• Sum of probabilities of all elementary events = 1

5. Addition Theorem on Probability

• If A and B are two events, then: P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
• If A and B are mutually exclusive: P(A ∪ B) = P(A) + P(B)

6. Conditional Probability

The probability of event A occurring given that event B has already occurred is called conditional probability, denoted by P(A|B):

P(A|B) = P(A ∩ B) / P(B),   provided P(B) ≠ 0

7. Independent and Dependent Events

• Independent Events: Occurrence of one does not affect the other.
• Dependent Events: Occurrence of one affects the probability of the other.

8. Multiplication Theorem on Probability

• For independent events A and B: P(A ∩ B) = P(A) × P(B)
• For dependent events: P(A ∩ B) = P(A) × P(B|A)

9. Bayes' Theorem (Statement Only)

Bayes' theorem relates the conditional and marginal probabilities of random events. It is used to update the probability of an event based on new information.

10. Examples

• Example 1: What is the probability of getting a head when tossing a fair coin?
  Solution: Number of favourable outcomes = 1 (head), Total outcomes = 2 (head, tail)
  P(E) = 1/2
• Example 2: What is the probability of drawing a red card from a standard deck of 52 cards?
  Solution: Number of red cards = 26, Total cards = 52
  P(E) = 26/52 = 1/2
• Example 3: If two dice are thrown, what is the probability that the sum is 7?
  Solution: Favourable outcomes = 6, Total outcomes = 36
  P(E) = 6/36 = 1/6

11. Practice Questions

1. A bag contains 3 red and 2 blue balls. What is the probability of drawing a blue ball?
2. If a card is drawn from a deck, what is the probability that it is a king?
3. Two coins are tossed. What is the probability of getting at least one head?
4. What is the probability of getting a number greater than 4 when a die is thrown?

12. Summary

• Probability quantifies uncertainty and helps in decision making.
• Probability values range from 0 (impossible) to 1 (certain).
• Key concepts: random experiment, sample space, event, mutually exclusive, independent events.
• Important formulas: P(E) = m/n, Addition and Multiplication theorems, Conditional probability.