Sample Space, Theoretical Probability, and Experimental Probability

Sample Space

  • Flipping two coins results in four possible outcomes: heads-heads, heads-tails, tails-heads, and tails-tails.

Combined Events

  • Flipping two coins and rolling a six-sided die involves three events.

  • To find the total number of outcomes, multiply the possibilities for each event.

  • Flipping two coins has 2 outcomes each, and rolling a six-sided die has 6 outcomes.

  • Total outcomes: 2×2×6=242 \times 2 \times 6 = 24

Listing All Possible Outcomes

  • List all combinations: (Heads, Heads, 1), (Heads, Heads, 2), …, (Heads, Heads, 6), (Heads, Tails, 1), …, (Tails, Tails, 6).

  • There should be 24 different outcomes in total.

Definition of Sample Space

  • A sample space lists all possible outcomes of an event.

Theoretical Probability

  • Theoretical probability represents the odds of an event occurring.

  • Example: Rolling a one on a six-sided die has a probability of 1/6.

  • Flipping tails on a coin has a theoretical probability of 1/2.

  • It represents what should happen, not necessarily what will happen in practice.

Experimental Probability

  • Experimental probability may differ from theoretical probability due to random chance.

  • Flipping a coin might not perfectly result in 50% heads and 50% tails in a real-world experiment.

Formula for Theoretical Probability

  • Theoretical Probability = (Number of Favorable Outcomes) / (Total Number of Outcomes)

  • Probability of rolling a 3: P(3)=16P(3) = \frac{1}{6}

  • Probability of flipping tails: P(Tails)=12P(Tails) = \frac{1}{2}

  • Notation: P(A)P(A) represents the probability of event A.

Example: Quiz with True/False Questions

  • A student guesses on four true/false questions.

  • Each question has two possible outcomes (correct or incorrect).

  • Total number of outcomes: 2×2×2×2=162 \times 2 \times 2 \times 2 = 16

Sample Space for the Quiz

  • Listing all possible outcomes (C = Correct, I = Incorrect):

    • All correct: CCCC

    • Three correct: CCCI, CCIC, CICC, ICCC

    • One correct: CIIII, ICIII, IICII, IIICI

    • All incorrect: IIII

Determining Outcomes with Two Correct Answers

  • There are 6 outcomes with exactly two correct answers.

    • CCII, CICI, CIIC, ICCI, ICIC, IICC

Calculating Probability

  • Probability of guessing exactly two correct answers: P(2 correct)=616=0.375=37.5%P(2 \text{ correct}) = \frac{6}{16} = 0.375 = 37.5\%

  • Probability of guessing all correct: P(4 correct)=116P(4 \text{ correct}) = \frac{1}{16}

Example: Flipping Tails and Rolling a Four

  • Probability of flipping tails and rolling a four on a six-sided die.

  • Total outcomes: 12

  • Only one outcome matches (Tails, 4).

  • P(Tails and 4)=112=0.083≈8.3%P(\text{Tails and 4}) = \frac{1}{12} = 0.083 \approx 8.3\%

Complements

  • The complement of an event is the opposite of that event.

  • Notation: P(A‾)P(\overline{A}) represents the probability of the complement of event A.

  • Formula: P(A‾)=1−P(A)P(\overline{A}) = 1 - P(A)

Example: Rolling Two Dice

  • Total possible outcomes when rolling two dice: 36.

  • Finding the probability that the sum is not six using the complement.

  • Find the probability that the sum is six. (5/36)

  • P(sum is not 6)=1−536=0.861≈86.1%P(\text{sum is not 6}) = 1 - \frac{5}{36} = 0.861 \approx 86.1\%

Inequality Complements

  • The complement of "less than or equal to" is "greater than."

  • The complement of "greater than or equal to" is "less than."

Example: Sum is Less Than or Equal to Nine

  • Find the probability that the sum is less than or equal to nine.

  • Find the complement: probability that the sum is greater than nine.

  • There are 6 outcomes where the sum is greater than nine.

  • P(sum≤9)=1−636=0.833≈83.3%P(\text{sum} \le 9) = 1 - \frac{6}{36} = 0.833 \approx 83.3\%

Example: Sum is Greater Than Three

  • Find the probability that the sum is greater than three using the complement.

  • Complement: probability that the sum is less than or equal to three.

  • P(\text{sum} > 3) = 1 - P(\text{sum} \le 3) = 1 - \frac{3}{36} = 0.9167 \approx 91.67\%

Experimental Probability

  • Experimental probability is based on actual experiments and may differ from theoretical probability.

  • Formula: Experimental Probability = (Number of times the event occurs) / (Total number of trials)

Spinner Example

  • A spinner with four equal sections is spun 20 times.

  • Theoretical probability of landing on one color: 1/4 = 0.25 = 25%.

  • Compare theoretical probability with experimental results to find the color with the same probabilities.

  • Red: 5/20 = 25%. The experimental probability of red matches the theoretical probability.