Chapter 6: Probability

6.1 Simple Probability

Probability is the measure of how likely an event is to occur. It is expressed as a number between 0 (impossible) and 1 (certain).

Key Concepts:

• Experiment:

  A process that leads to an outcome.

• Sample Space (S):

  The set of all possible outcomes of an experiment.

• Event (E):

  A subset of the sample space representing specific outcomes.

• Outcome:

  A single possible result of an experiment.
  • Example: Rolling a die can result in outcomes \(1,2,3,4,5,\) or \(6\).
  • Example: Flipping a coin has two possible outcomes: **Heads** or **Tails**.

• Outcome Space (Sample Space, S):

  The set of all possible outcomes of an experiment.
  • Example: The sample space for rolling a six-sided die is: \[ S = \{1,2,3,4,5,6\}. \]
  • Example: The sample space for flipping two coins is: \[ S = \{\text{HH, HT, TH, TT}\}. \]

• Probability of an Event:

  The probability of an event occurring is given by: \[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}. \]

• Complement of an Event:

  The probability of an event not occurring is: \[ P(E') = 1 - P(E). \]

Examples:

Example 1: Finding the Probability of Rolling a Die

Solution:

Step 1: Identify the sample space. \\ The sample space for rolling a die is: \[ S = \{1,2,3,4,5,6\}. \] Step 2: Find the probability of rolling a 4. \[ P(4) = \frac{1}{6}. \] Thus, the probability of rolling a 4 is **\(\frac{1}{6}\)**.

Example 2: Probability of Drawing a Red Card from a Deck of 52 Cards

Solution:

Step 1: Identify the total outcomes. \\ A standard deck has 52 cards, and half (26) are red. Step 2: Compute the probability. \[ P(\text{Red Card}) = \frac{26}{52} = \frac{1}{2}. \] Thus, the probability of drawing a red card is **\(\frac{1}{2}\)**.