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Probability

• Probability: A field of Mathematics that deals with chances.

Experiment

• Experiment: An activity in which results cannot be predicted with certainty.

Outcome

• Outcome: The result of an experiment.

Sample Space

• Sample Space: The set that contains all the possible outcomes of the experiment.

  • Example for rolling a die: S = {1, 2, 3, 4, 5, 6}

  • Example for tossing a coin: S = {Head, Tail}

Probability of an Event

• Formula: P(event) = n(event) / n(sample space)

  • Where:

    • n(event) = Number of outcomes of the event

    • n(sample space) = Total number of possible outcomes

Example 1: Coin Toss

• A coin is tossed.

  • Find the probability of getting a head:

    • S = {Head, Tail}

    • P(head) = n(head) / n(sample space)

      • n(head) = 1

      • n(sample space) = 2

    • Therefore, P(head) = 1/2

Example 2: Card Drawing

• a. Probability of picking a black card at random from a standard deck of 52 cards.

• b. Probability of picking a face card at random from a standard deck of 52 cards.

• c. Probability of not picking a face card at random from a standard deck of 52 cards.

Example 3: Rolling a Die

• a. Probability of rolling a 3:

  • P(a 3) = n(a 3) / n(sample space)

  • n(sample space) = 6

  • Therefore, P(a 3) = 1/6

• b. Probability of rolling an even number:

  • P(even) = n(even) / n(sample space)

  • n(even) = 3 (2, 4, 6)

  • Therefore, P(even) = 3/6 = 1/2

• c. Probability of rolling a zero:

  • P(zero) = n(zero) / n(sample space)

  • n(zero) = 0

  • Therefore, P(zero) = 0/6 = 0

• d. Probability of rolling a number between 0 and 7.

Tossing a Coin Twice

• Possible outcomes:

  • Head and Head

  • Head and Tail

  • Tail and Head

  • Tail and Tail

• Sample Space Number of Tails:

  • TT = 2

  • TH = 1

  • HT = 1

  • HH = 0

  • Random Variable: X = {0, 1, 2}

Random Variable

• Definition: A set whose elements are the numbers assigned to the outcomes of an experiment.

  • Examples include:

    • Tossing a coin three times

    • Rolling a die twice

    • Drawing two balls in a box

Example 1: Random Variable for Coin Toss

• Let Y represent the number of tails that occur when three coins are tossed.

• Possible outcomes with respect to tails:

  • HHH -> 0 tails

  • HHT -> 1 tail

  • HTH -> 1 tail

  • HTT -> 2 tails

  • THH -> 1 tail

  • THT -> 2 tails

  • TTH -> 2 tails

  • TTT -> 3 tails

Example 2: Drawing Balls

• Situation: Two balls are drawn in succession without replacement from a box containing 5 red balls and 6 blue balls.

• Let Z represent the number of blue balls drawn:

  • Possible outcomes: RR -> 0 Blue, RB -> 1 Blue, BR -> 1 Blue, BB -> 2 Blue

  • Z = {0, 1, 2}

Random Variable Examples

• Example 3:

  • X: Number of even number outcomes in a roll of a die.

  • Y: Weight (in mg) of a powder that does not exceed 80 mg.

  • Z: Number of heads in 4 flips of a coin.

  • A: Length (in cm) of a shoelace that is not longer than 2 meters.

  • B: Scores of a student in a 10-item test.

  • C: Probability (in %) of raining today.

  • G: Number of typhoons that pass through PAS in a year.

  • H: Number of pages a Statistics book has.

Types of Random Variables

• Discrete Random Variable:

  • Takes on a finite number of distinct values.

  • Example: Number of students inside the classroom.

• Continuous Random Variable:

  • Takes an infinitely uncountable number of possible values, typically measurable quantities.

  • Example: Distance of the faculty room and the classroom.