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Mathematics Class 11 - Probability Notes

Comprehensive study notes for Class 11 - Probability olympiad preparation

Probability

Welcome to the chapter on Probability for Class 11. In this chapter, you will learn about the concept of probability, types of events, and how to calculate the probability of different events. By the end of this chapter, you will be able to solve problems involving probability and understand its applications in real life.

Key Concepts

  • Experiment: An action or process that leads to one or more outcomes.
  • Sample Space (S): The set of all possible outcomes of an experiment.
  • Event: A subset of the sample space. It may contain one or more outcomes.
  • Favourable Outcomes: The outcomes which make an event happen.
  • Probability of an Event (E): P(E) = Number of favourable outcomes / Total number of outcomes

Types of Events

  • Simple Event: An event with only one outcome.
  • Compound Event: An event with more than one outcome.
  • Impossible Event: An event that cannot happen (probability = 0).
  • Certain Event: An event that is sure to happen (probability = 1).
  • Mutually Exclusive Events: Events that cannot happen at the same time.
  • Exhaustive Events: All possible outcomes together.

Classical Definition of Probability

If there are n equally likely, mutually exclusive, and exhaustive outcomes of an experiment, and m of them are favourable to an event E, then:

P(E) = m / n

Properties of Probability

  • The probability of any event lies between 0 and 1: 0 ≤ P(E) ≤ 1
  • The sum of probabilities of all possible outcomes is 1.
  • The probability of an impossible event is 0.
  • The probability of a certain event is 1.

Examples

  • Example 1: What is the probability of getting a head when tossing a fair coin?
    Sample space S = {Head, Tail} (n = 2)
    Favourable outcomes = 1 (Head)
    P(E) = 1/2
  • Example 2: What is the probability of getting a 3 when rolling a fair die?
    Sample space S = {1, 2, 3, 4, 5, 6} (n = 6)
    Favourable outcomes = 1 (3)
    P(E) = 1/6
  • Example 3: What is the probability of getting an even number when rolling a die?
    Even numbers = {2, 4, 6} (favourable outcomes = 3)
    P(E) = 3/6 = 1/2

Applications of Probability

  • Games of chance (cards, dice, coins)
  • Weather forecasting
  • Insurance and risk assessment
  • Genetics and biology
  • Decision making in daily life

Practice Questions

  1. A bag contains 5 red balls and 3 blue balls. What is the probability of picking a blue ball?
  2. What is the probability of getting a number less than 4 when rolling a die?
  3. If a coin is tossed twice, what is the probability of getting at least one head?
  4. What is the probability of drawing an ace from a standard deck of 52 cards?
  5. If an event is impossible, what is its probability?

Challenge Yourself

  • If two dice are thrown together, what is the probability that the sum is 7?
  • A card is drawn at random from a deck of 52 cards. What is the probability that it is a face card?

Did You Know?

  • Probability is used in computer science, economics, and many other fields.
  • The word "probability" comes from the Latin word "probabilitas," meaning likelihood.

Glossary

  • Experiment: An action with uncertain results.
  • Sample Space: All possible outcomes of an experiment.
  • Event: One or more outcomes of an experiment.
  • Mutually Exclusive: Events that cannot happen together.

Answers to Practice Questions

  1. P(blue) = 3/(5+3) = 3/8
  2. Numbers less than 4: {1, 2, 3} ⇒ P = 3/6 = 1/2
  3. Outcomes: HH, HT, TH, TT. At least one head: HH, HT, TH ⇒ P = 3/4
  4. There are 4 aces in 52 cards ⇒ P = 4/52 = 1/13
  5. 0

Practice probability problems to strengthen your understanding and apply it in real life!

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