1. Introduction to Probability

Probability is the branch of mathematics that measures uncertainty. In our daily lives, we constantly encounter situations where the outcome is not certain—will it rain tomorrow? Will my favorite team win the match? Probability provides a precise way to quantify the chance or likelihood of each possible outcome. 🎲

For example, a weather forecast stating a "70% chance of rain" is a probability. It doesn't guarantee rain, but it tells us that rain is a more likely outcome than not. The theory of probability is a fundamental tool in many fields, including science, economics, engineering, and finance.

2. Random Experiments

An experiment is any action or process that results in an observable outcome. A random experiment is a special type of experiment with two key properties:

  1. It has more than one possible outcome.
  2. It is not possible to predict the outcome in advance with certainty.

Examples of Random Experiments:

  • Tossing a coin (Outcomes: Head, Tail)
  • Rolling a die (Outcomes: 1, 2, 3, 4, 5, 6)
  • Drawing a card from a standard deck of 52 cards.

Non-Example: Dropping a stone from your hand is an experiment, but it is not a random experiment. There is only one certain outcome: the stone will fall due to gravity.

3. Sample Space

The sample space, denoted by SS, is the set of all possible outcomes of a random experiment. Each individual outcome within the sample space is called a sample point.

Examples:

  • Experiment: Tossing a single coin.

    • Sample Space (S): {Head, Tail} or simply {H, T}.
    • The number of sample points is n(S)=2n(S) = 2.

  • Experiment: Rolling a single die.

    • Sample Space (S): {1, 2, 3, 4, 5, 6}.
    • n(S)=6n(S) = 6.

  • Experiment: Tossing two coins simultaneously.

    • Sample Space (S): {HH, HT, TH, TT}. (HT means Head on the first coin, Tail on the second).
    • n(S)=4n(S) = 4.

  • Experiment: Rolling two dice simultaneously.

    • The outcome is an ordered pair. For example, (1, 2) is different from (2, 1).
    • Sample Space (S): {(1,1), (1,2), (1,3), ..., (6,6)}.
    • n(S)=6×6=36n(S) = 6 \times 6 = 36.

4. Events

An event is a specific outcome or a collection of outcomes of an experiment. Formally, an event (denoted by a capital letter like E) is any subset of the sample space SS. An event is said to occur if the result of the experiment is a sample point that belongs to the event set.

Example:

  • Experiment: Rolling two dice. The sample space SS has 36 outcomes.
  • Event A: "The sum of the numbers is 5."
    • Event Set A: {(1,4), (2,3), (3,2), (4,1)}. This is a subset of SS.
    • If you roll the dice and get (2,3), we say that Event A has occurred.
  • Event B: "Getting a doublet (both dice show the same number)."
    • Event Set B: {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}.

Types of Events

  • Impossible Event: An event that cannot happen. Its set is the empty set, ∅\emptyset.

    • Example: The event of getting a '7' when rolling a single die. The event set is { }, or ∅\emptyset.
    • Example: The event of getting a sum of 13 when rolling two dice.

  • Sure Event: An event that is certain to happen. Its set is the entire sample space, SS.

    • Example: The event of getting a number less than 7 when rolling a single die. The event set is {1, 2, 3, 4, 5, 6}, which is SS.
    • Example: The event of getting a sum between 2 and 12 (inclusive) when rolling two dice.

  • Simple (or Elementary) Event: An event consisting of only a single sample point.

    • Example: The event of getting exactly '3' when rolling a die. The event set is E={3}E = \{3\}.
    • Example: The event of getting two heads when tossing two coins. The event set is E={HH}E = \{HH\}.

  • Compound Event: An event consisting of more than one sample point.

    • Example: The event of getting an odd number when rolling a die. The event set is E={1,3,5}E = \{1, 3, 5\}.
    • Example: The event of getting at least one head when tossing two coins. The event set is E={HT,TH,HH}E = \{HT, TH, HH\}.