Exam Review: Probability and Statistical Concepts

Probability and Variance

Spinner Game Example

Game Setup

  • Cost to play: $7.00\$7.00

  • Spinner Outcomes: 2,4,6,8,102, 4, 6, 8, 10

  • Probabilities: Each outcome has an even probability of 15\frac{1}{5}. The term "inclusive" means the boundaries are included.

Expected Value (Average Profit)

  • Average Spin (Revenue):
    2+4+6+8+105=305=6\frac{2 + 4 + 6 + 8 + 10}{5} = \frac{30}{5} = 6
    The average revenue per spin is $6.00\$6.00.

  • Average Profit (Expected Value):
    E(X)=Average Revenue−Average Cost=$6−$7=−$1E(X) = \text{Average Revenue} - \text{Average Cost} = \$6 - \$7 = -\$1
    The expected value is −$1.00-\$1.00, meaning on average, a player loses $1.00\$1.00 per game.

Detailed Profit & Expected Value Calculation

  • Profit per outcome (Outcome - Cost):

    • 2−7=−52 - 7 = -5

    • 4−7=−34 - 7 = -3

    • 6−7=−16 - 7 = -1

    • 8−7=18 - 7 = 1

    • 10−7=310 - 7 = 3

  • Calculation of Expected Value using individual profits and probabilities:
    E(X)=∑<em>i=1nP(x</em>i)⋅ProfitiE(X) = \sum<em>{i=1}^{n} P(x</em>i) \cdot \text{Profit}_i
    E(X)=15(−5)+15(−3)+15(−1)+15(1)+15(3)E(X) = \frac{1}{5}(-5) + \frac{1}{5}(-3) + \frac{1}{5}(-1) + \frac{1}{5}(1) + \frac{1}{5}(3)
    E(X)=15(−5−3−1+1+3)=15(−5)=−1E(X) = \frac{1}{5}(-5 - 3 - 1 + 1 + 3) = \frac{1}{5}(-5) = -1
    This confirms the expected value of −$1.00-\$1.00. The presenter notes that canceling out positive and negative terms (e.g., −3-3 and +3+3) can minimize math errors.

Variance and Standard Deviation

  • Variance Formula: Var(X)=E(X2)−(E(X))2Var(X) = E(X^2) - (E(X))^2

  • Calculate squared profits for E(X2)E(X^2):

    • (−5)2=25(-5)^2 = 25

    • (−3)2=9(-3)^2 = 9

    • (−1)2=1(-1)^2 = 1

    • (1)2=1(1)^2 = 1

    • (3)2=9(3)^2 = 9

  • Calculate E(X2)E(X^2):
    E(X2)=15(25)+15(9)+15(1)+15(1)+15(9)E(X^2) = \frac{1}{5}(25) + \frac{1}{5}(9) + \frac{1}{5}(1) + \frac{1}{5}(1) + \frac{1}{5}(9)
    E(X2)=15(25+9+1+1+9)=15(45)=9E(X^2) = \frac{1}{5}(25 + 9 + 1 + 1 + 9) = \frac{1}{5}(45) = 9

  • Calculate Variance (Var(X)Var(X)):
    Var(X)=9−(−1)2=9−1=8Var(X) = 9 - (-1)^2 = 9 - 1 = 8
    It's crucial to always square the expected value, even if it's negative, before subtracting, as variance can never be negative.

  • Standard Deviation Formula: σ=Var(X)\sigma = \sqrt{Var(X)}

  • Calculate Standard Deviation (σ\sigma):
    σ=8≈2.8284\sigma = \sqrt{8} \approx 2.8284
    (Rounded to four decimal places as requested.)

Significance of Variance and Standard Deviation

  • Standard Deviation: Often more directly relevant for day-to-day understanding of data spread.

  • Variance: Useful for certain mathematical manipulations, such as adding variances of independent populations (e.g., combining the variances of scores from two separate tests).

Types of Probability

Theoretical Probability

  • Definition: Probability derived from logical reasoning and known conditions or assumptions, without conducting experiments.

  • Example: The theoretical probability of rolling a six on a standard six-sided die is 16\frac{1}{6}.

  • Application in the course: Previous examples, like the spinner game, have primarily used theoretical probability.

Empirical Probability

  • Definition: Probability determined by observing the outcomes of actual experiments or collecting data.

  • Empirical Method/Scientific Method: A formal approach to