Probability - Class 11 Mathematics

Introduction

• In this lecture, we will study the important principles and applications of probability.
• We will see how to solve probability questions.

What is Probability?

• Probability: It indicates the likelihood of an event occurring.
• Example: The probability of rain, the probability of a specific outcome in a dice roll in Ludo.
• Classical Probability: The number of favorable outcomes divided by the total number of possible outcomes.

Definitions and Important Concepts

• Experiment: Any process in which the potential outcome is unknown.
• Random Experiment: An experiment whose results are uncertain. Example: Tossing a coin, throwing a dice.
• Sample Space: The set of all possible results. Example: On throwing a dice, the possible outcomes are 1, 2, 3, 4, 5, 6.
• Event: A subset of the sample space. Example: An event on a dice could be "getting an even number".
• Impossible Event: An event that cannot occur. Example: Getting a 7 on a dice.
• Certain Event: An event that is sure to occur. Example: Getting a number between 1 and 6 on a dice.
• Simple Event: Which has only one sample point in the sample space.
• Compound Event: Which has more than one sample point.
• Mutually Exclusive Events: Events that cannot happen simultaneously. Example: Getting heads and tails in one coin toss.
• Exhaustive Events: Events that together cover the entire sample space.

Rules of Probability

• Probability of event A:

  P(A) = (Number of favorable outcomes for event A) / (Total number of possible outcomes)

• The probability of an event is always between 0 and 1.

• The probability of the sample space is always 1.

Types of Probability

• Classical Probability: Counting equally likely outcomes.
• Geometric Probability: The likelihood of events in a continuum.
  • Example: Choosing a point.

Probability of Combined Events

• P(A or B) : The probability of A or B or both happening.

  P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

• P(A and B): The probability of both A and B happening.

  P(A ∩ B) = P(A) * P(B) (if independent)

• P(A') : The probability of the non-occurrence of A.

  P(A') = 1 – P(A)

Important Examples

• Coin Tossing:

  • Possible outcomes: H, T
  • P(H) = 1/2
  • P(T) = 1/2

• Throwing a Dice:

  • Possible outcomes: 1, 2, 3, 4, 5, 6
  • P(getting a number) = 1/6
  • P(getting an even number) = 3/6 = 1/2

• Example of a Compound Event:

  • Throwing two dice
  • Sample Space: (1,1), (1,2), ..., (6,6)
  • P(both even numbers) = 3/6 * 3/6

Supplementary Questions

• Review vocabulary and theorems.
• Identify and list sample spaces and possible events.
• Solve questions based on different types of probability.

Conclusion

• Probability is an important part of mathematics that helps us understand the likelihood of various events happening.
• Students gain the ability to solve math problems through these principles and practical examples.

For more questions and practice, work from the textbook and ask teachers or classmates for any doubts.