Probability

Probability is a branch of mathematics that deals with measuring the likelihood of an event to occur. It is widely used in daily life, science, engineering, and statistics to predict outcomes and make informed decisions.

1. Introduction to Probability

• Probability quantifies uncertainty.
• The value of probability ranges from 0 (impossible event) to 1 (certain event).
• Examples: Tossing a coin, rolling a die, drawing a card from a deck.

2. Basic Terms

• Experiment: An action or process that leads to one or more outcomes (e.g., tossing a coin).
• Sample Space (S): The set of all possible outcomes (e.g., for a coin toss: S = {Head, Tail}).
• Event: A subset of the sample space (e.g., getting a Head).
• Favourable Outcomes: Outcomes that satisfy the event.

3. Classical Definition of Probability

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

4. Examples

• Example 1: What is the probability of getting a 3 when a fair die is rolled?
  Solution: There are 6 possible outcomes (1 to 6). Only one outcome is 3.
  P(getting 3) = 1/6
• Example 2: What is the probability of getting a head when a coin is tossed?
  Solution: There are 2 possible outcomes (Head, Tail).
  P(getting Head) = 1/2
• Example 3: What is the probability of drawing a red card from a standard deck of 52 cards?
  Solution: There are 26 red cards.
  P(red card) = 26/52 = 1/2

5. Properties of Probability

• 0 ≤ P(E) ≤ 1 for any event E.
• P(Sample Space) = 1
• P(Impossible Event) = 0
• P(Not E) = 1 – P(E)

6. Types of Events

• Sure Event: An event that is certain to happen (probability = 1).
• Impossible Event: An event that cannot happen (probability = 0).
• Equally Likely Events: Events that have the same chance of occurring.
• Mutually Exclusive Events: Events that cannot happen at the same time.
• Exhaustive Events: All possible outcomes together.

7. Practice Questions

1. A card is drawn at random from a deck of 52 cards. What is the probability that it is a king?
2. What is the probability of getting an even number when a die is rolled?
3. If a coin is tossed twice, what is the probability of getting at least one head?
4. What is the probability of drawing a black ace from a deck of cards?

8. Common Mistakes

• Not listing all possible outcomes in the sample space.
• Assuming outcomes are equally likely when they are not.
• Forgetting to reduce fractions to their simplest form.

9. Summary

• Probability measures the chance of an event occurring.
• It is always between 0 and 1.
• Probability = (Number of favourable outcomes) / (Total number of outcomes).
• Understanding probability helps in making predictions and informed decisions.