Probability

1. Introduction to Probability

Probability is the branch of mathematics that deals with the likelihood or chance of different outcomes. It helps us predict how likely an event is to happen.

• Probability is used in games, weather forecasts, insurance, and many real-life situations.
• It is a measure between 0 (impossible event) and 1 (certain event).

2. Basic Terms

• Experiment: An action or process that leads to one or more outcomes (e.g., tossing a coin).
• Trial: Each repetition of an experiment.
• Outcome: The result of a single trial (e.g., getting heads in a coin toss).
• Sample Space (S): The set of all possible outcomes (e.g., for a coin toss: S = {Heads, Tails}).
• Event: One or more outcomes of an experiment (e.g., getting an even number when rolling a die).

3. Calculating Probability

The probability of an event A is given by:

Probability is always a number between 0 and 1.

4. Examples

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

5. Important Points

• Probability of an impossible event is 0.
• Probability of a certain event is 1.
• The sum of probabilities of all possible outcomes of an experiment is 1.
• Probabilities are always between 0 and 1 (inclusive).

6. Common Mistakes

• Not listing all possible outcomes in the sample space.
• Counting the same outcome more than once.
• Forgetting that probability cannot be negative or greater than 1.

7. Practice Questions

1. What is the probability of getting a tail when tossing a coin?
2. What is the probability of getting a number greater than 4 when rolling a die?
3. A bag contains 3 red balls and 2 blue balls. What is the probability of picking a blue ball?
4. What is the probability of getting an odd number when rolling a die?
5. What is the probability of drawing a king from a standard deck of 52 cards?

8. Summary

• Probability measures how likely an event is to occur.
• P(A) = Number of favourable outcomes / Total number of possible outcomes.
• Probability values range from 0 to 1.
• Understanding probability helps in making predictions and informed decisions.