lesson 10 basic probability

  • Coin Toss:

    • Outcomes of a coin toss: Heads (H) or Tails (T)

    • Each toss gives one outcome.

  • Rolling a Die:

    • A die has six faces numbered 1 to 6.

    • The outcomes when rolling a die: {1, 2, 3, 4, 5, 6}.

    • Each roll yields one outcome.

  • Tossing a Coin Twice:

    • Possible outcomes when tossing a coin twice:

    • HH (Heads, Heads)

    • TT (Tails, Tails)

    • HT (Heads, Tails)

    • TH (Tails, Heads)

    • Total of 4 outcomes.

  • Creating a Table for Outcomes:

    • Table method for organizing outcomes.

    • A two by two table can be created for two coin tosses.

    • Rows represent the first coin toss outcomes (H/T).

    • Columns represent the second coin toss outcomes (H/T).

    • Example table arrangement:
      | | H | T |
      |-------|-------|-------|
      | H | HH | HT |
      | T | TH | TT |
      This produces four unique outcomes.

  • Rolling a Pair of Dice:

    • Outcomes when rolling two dice can also be organized in a table.

    • A table showing all combinations like this:
      | | 1 | 2 | 3 | 4 | 5 | 6 |
      |-------|-------|-------|-------|-------|-------|-------|
      | 1 | 1,1 | 1,2 | 1,3 | 1,4 | 1,5 | 1,6 |
      | 2 | 2,1 | 2,2 | 2,3 | 2,4 | 2,5 | 2,6 |
      | 3 | 3,1 | 3,2 | 3,3 | 3,4 | 3,5 | 3,6 |
      | 4 | 4,1 | 4,2 | 4,3 | 4,4 | 4,5 | 4,6 |
      | 5 | 5,1 | 5,2 | 5,3 | 5,4 | 5,5 | 5,6 |
      | 6 | 6,1 | 6,2 | 6,3 | 6,4 | 6,5 | 6,6 |

    • Total outcomes: 36 (since 6imes6=366 imes 6 = 36).

  • Sample Space:

    • The sample space is defined as the set of all possible outcomes for each experiment.

    • In the example of rolling dice, the sample space includes all 36 outcomes listed in the table.

  • Events:

    • An event is defined as a collection of one or more outcomes from an experiment.

    • Events can have different notations (e.g., Event A for rolling a seven).

    • Example for event A (rolling a seven): Outcomes are {1,6}, {2,5}, {3,4}, {4,3}, {5,2}, {6,1}.

    • Event B (rolling a twelve): Only the outcome {6,6}.

  • Simple vs Compound Events:

    • A simple event involves only one outcome, while a compound event involves multiple outcomes.

  • Probability Definition:

    • Probability is a numerical representation of the likelihood of an event occurring.

  • Calculating Probability:

    • The classical approach to probability calculates the probability of an event as:
      P(A)=n(A)n(S)P(A) = \frac{n(A)}{n(S)}
      where n(A)n(A) is the number of favorable outcomes, and n(S)n(S) is the total number of outcomes in the sample space.

    • Example: If event A is rolling a six, then P(A)=636=16P(A) = \frac{6}{36} = \frac{1}{6}.

    • Probability values are typically expressed as decimals, e.g., P(A)=0.1667P(A) = 0.1667 (to four decimal places).

  • Key Points:

    • Be precise in wording events for accurate probability calculation.

    • Understanding the distinction between events and outcomes helps in better comprehending probability problems.