Discrete Probability Distributions and Expected Value

Counting Formulas Review: Permutations vs. Combinations

• Primary Formulas in Counting Outcomes:

  • Permutations ($nPr$) and Combinations ($nCr$) are the standard calculation methods for determining the total number of ways events can occur.

• Permutations ($nPr$):

  • Core Principle: Order matters.

  • Mnemonic: Permutation corresponds to Position (position, range, sequence, order, rank, or arrangement).

  • Key Trigger Words: Position, arrange, sequence, order, rank.

  • Practical Example: Entering test scores into a grade book requires sorting student grades in alphabetical order. Because the scores must be typed in a specific sequential order corresponding to student names, the sequence matters ($nPr$).

• Combinations ($nCr$):

  • Core Principle: Order does NOT matter.

  • Mnemonic: Combinations are for Committees.

  • Key Concept: On a committee, members form an unordered group ("blob of people"). There are no unique positions, ranks, sequences, or hierarchical levels among members.

Course Schedule and Examination Details

• Coverage: The upcoming examination covers Chapter 4 and Chapter 5.

• Weekly Schedule:

  • Tuesday (Day 1): Section 5.1 (Discrete Probability Distributions).

  • Thursday (Day 2): Section 5.2.

  • Next Tuesday (1 Week from Today): Comprehensive Test Review Day. The test review material is published early for advance practice.

  • Next Thursday (1 Week from Thursday): In-class Chapter 4 & 5 Examination.

Introduction to Discrete Probability Distributions

• Conceptual Continuity: Chapter 5 does not introduce entirely new arithmetic concepts; rather, it applies existing methods (calculating means, standard deviations, and probabilities) using formal mathematical notation.

• Notation Shift (Chapter 4 vs. Chapter 5):

  • Chapter 4 Approach: Rolling a single fair six-sided die. Sample space is $\{1, 2, 3, 4, 5, 6\}$, consisting of $6$ equally likely outcomes. The probability of rolling a $2$ is expressed directly as:         P(2) = \n\frac{1}{6}

  • Chapter 5 Approach: Define a random variable $X$ equal to the result of the die roll. The probability of rolling a $2$ is expressed in function notation as:         P(X = 2) = \n\frac{1}{6}

• Probability Distribution Table:

  • Constructed similarly to relative frequency distribution tables.

  • The left column lists every possible numerical value of the random variable $X$.

  • The right column lists the associated probability $P(X)$ for each corresponding outcome.

  • Example Table (Fair Six-Sided Die Roll):

    • X = 1 \implies P(X) = \n\frac{1}{6}

    • X = 2 \implies P(X) = \n\frac{1}{6}

    • X = 3 \implies P(X) = \n\frac{1}{6}

    • X = 4 \implies P(X) = \n\frac{1}{6}

    • X = 5 \implies P(X) = \n\frac{1}{6}

    • X = 6 \implies P(X) = \n\frac{1}{6}

Random Variables: Definitions and Classifications

• Random Variable ($X$):

  • Definition: A variable, typically represented by $X$, that has a single numerical value determined by chance for each outcome of a procedure.

  • Property: Numerical values occur purely due to chance (e.g., rolling a die produces an outcome from $\{1, 2, 3, 4, 5, 6\}$ at random).

• Probability Distribution:

  • Definition: A description (in table, formula, or graphical form) that gives the probability for each value of the random variable.

• Classifications of Random Variables:

  • Discrete Random Variable:

    • Has a finite or countable list of numerical outcomes.

    • Consists of counting numbers without intermediate fractional steps.

    • Example: Rolling a die yields $X \in \{1, 2, 3, 4, 5, 6\}$. Impossible outcomes include $7$ or $8.2$

  • Continuous Random Variable:

    • Has infinitely many values covering a continuous numerical scale without gaps or interruptions.

    • Example: Commute time driving from St. Pete every morning ranging between $40$ and $60$ minutes. Outcomes can include $40.2\,\text{minutes}$, $41\,\text{minutes}$, $41.5\,\text{minutes}$, or $41.57\,\text{minutes}$.

  • Scope Note: Sections 5.1 and 5.2 focus exclusively on discrete random variables. Continuous random variables are addressed in Chapter 6.

Requirements for a Valid Probability Distribution

• To confirm whether a given distribution is a valid probability distribution, all three of the following requirements must evaluate to YES:

  1. Numerical Random Variable: The random variable $X$ must represent quantitative numerical values. Non-numerical qualitative terms (e.g., colors like red/green/blue, or categories like "inappropriate attire") violate this rule.

  2. Sum of Probabilities Equals 1: Summing all individual probabilities across the sample space must yield exactly $1$:         \n\nsigmap P(X) = 1

  3. Valid Probability Range: Every individual probability $P(X)$ must fall inclusively between $0$ and $1$:         $0 \le P(X) \le 1$

• Evaluation of Proposed Distributions:

  • Case 1: Die Roll Distribution

    • $X \in \{1, 2, 3, 4, 5, 6\}$, each with P(X) = \n\frac{1}{6}.

    • Numerical $X$? Yes.

    • \n\nsigmap P(X) = \n\n\frac{6}{6} = 1? Yes.

    • $0 \le P(X) \le 1$? Yes (0 \le \n\frac{1}{6} \le 1).

    • Result: Valid probability distribution.

  • Case 2: Number of Heads in Two Coin Flips

    • Sample space: $\{\text{HH}, \text{HT}, \text{TH}, \text{TT}\}$.

    • Let $X$ = number of heads: $X \in \{0, 1, 2\}$.

    • $P(X = 0) = 0.25$ (from outcome $\text{TT}$, $1$ of $4$ outcomes).

    • $P(X = 1) = 0.50$ (from outcomes $\text{HT}, \text{TH}$, $2$ of $4$ outcomes).

    • $P(X = 2) = 0.25$ (from outcome $\text{HH}$, $1$ of $4$ outcomes).

    • Numerical $X$? Yes ($0, 1, 2$).

    • \n\nsigmap P(X) = 0.25 + 0.50 + 0.25 = 1.0? Yes.

    • $0 \le P(X) \le 1$? Yes.

    • Result: Valid probability distribution. Note that while underlying outcomes in the sample space are equally likely, the values of $X$ in the distribution table do not have to be equally likely ($0.25, 0.50, 0.25$).

  • Case 3: American Airlines Flights ($n = 5$ Flights Selected)

    • Let $X$ = number of flights landing on time: $X \in \{0, 1, 2, 3, 4, 5\}$.

    • Probabilities sum to $1.0$ and each individual probability is between $0$ and $1$.

    • Result: Valid probability distribution.

    • Variable Type: Discrete, because the values of $X$ represent countable whole numbers ($0, 1, 2, 3, 4, 5$).

  • Case 4: Inappropriate Attire Proposed Distribution

    • $X = \text{inappropriate attire}$.

    • Result: Invalid probability distribution.

    • Reasoning: $X$ is non-numerical/categorical, and the probabilities do not sum to $1$.

Probability Histograms

• Definition: A graphical representation of a probability distribution.

• Structure:

  • Identical in structure to relative frequency histograms.

  • Horizontal Scale: Displays the values of the numerical discrete random variable $X$.

  • Vertical Scale: Displays probabilities $P(X)$ rather than raw relative frequencies.

  • Example (Two Coin Flips): Bar heights correspond directly to values $P(0) = 0.25$, $P(1) = 0.50$, and $P(2) = 0.25$

Parameters of a Probability Distribution

• Population Classification: A probability distribution models the theoretical population of all possible outcomes of a procedure (e.g., all possible die rolls or flight arrival outcomes). Therefore, measures of center and variation are parameters, not sample statistics.

• Notation:

  • Population Mean: \n\nmu

  • Population Standard Deviation: \n\nsigma

• Mathematical Formulas:

  • Mean (\n\nmu):         \n\nmu = \n\nsigmap [X \times P(X)]

  • Standard Deviation (\n\nsigma):         \n\nsigma = \sqrt{\n\nsigmap [(X - \n\nmu)^2 \times P(X)]}

Calculating Mean and Standard Deviation Using Technology

• TI-83/84 Calculator Procedure (1-Var Stats):

  1. Press STAT $\rightarrow$ Edit...

  2. Enter values of the random variable $X$ into list L1 ($0, 1, 2, 3, 4, 5$).

  3. Enter corresponding probabilities $P(X)$ into list L2.

  4. Press STAT $\rightarrow$ CALC $\rightarrow$ Select 1: 1-Var Stats.

  5. Set List: to L1 (2nd $\rightarrow$ 1).

  6. Set FreqList: to L2 (2nd $\rightarrow$ 2) to apply probability weighting.

  7. Select Calculate.

  • Older Calculator Models: Screen displays 1-Var Stats. Type L1, L2 using the comma key (1-Var Stats L1, L2) and press ENTER.

• Detailed Example: American Airlines On-Time Flights ($n = 5$ Flights)

  • Random Variable $X \in \{0, 1, 2, 3, 4, 5\}$.

  • Probabilities for low outcomes: $P(0)$ is extremely small, $P(1) = 0.006$, $P(2)$ is small. Probabilities are heavily weighted toward higher outcomes ($X = 3, 4, 5$).

  • Intuition Check:

    • Expected Mean: Located near $3, 4, \text{or } 5$ due to heavy probability weights at the top end.

    • Expected Standard Deviation: Anticipated around $1\,\text{or } 2$ flights (typical distance of outcomes from the center).

  • Calculator Output:

    • Mean: \n\nmu = 4.0\,\text{on-time flights}.

    • Standard Deviation: \n\nsigma \approx 0.9\,\text{on-time flights}.

  • Practical Interpretation: In any random sample of $5$ flights, an average of $4.0$ flights are expected to be on time, with a standard deviation of $0.9$ flights (meaning $4$ on time and $1$ late is typical).

  • Identification of Unusual Events ($5\% = 0.05$ Cutoff Rule):

    • Getting $0$ on-time flights: Unusual ($P(0) < 0.05$).

    • Getting $1$ on-time flight ($P(1) = 0.006$): Unusual ($P(1) < 0.05$).

    • Getting $2$ on-time flights: Usual ($P(2) > 0.05$).

Expected Value and Its Applications

• Expected Value ($E$ or $E(X)$):

  • Definition: The expected value of a discrete random variable $X$ represents the theoretical mean value of outcomes resulting from infinitely repeated trials.

  • Mathematical Identity:         E = \n\nmu = \n\nsigmap [X \times P(X)]

  • Synonymous Terms: Expected Value of $X$ = $E(X)$ = Population Mean \n\nmu

• Hand Calculation Method vs. Calculator:

  • Formula Application Example (Two Coin Flips, Number of Heads):

    • $X \in \{0, 1, 2\}$ with $P(0) = 0.25$, $P(1) = 0.50$, $P(2) = 0.25$.

    • $E = (0 \times 0.25) + (1 \times 0.50) + (2 \times 0.25)$

    • $E = 0 + 0.50 + 0.50 = 1.0\,\text{head}$

  • Interpretation: Flipping two fair coins yields an expected value of $1.0\,\text{head}$.

Real-World Application: Expected Value in Gambling (Roulette vs. Craps)

• Scenario: A bettor enters the Golden Nugget casino in Las Vegas with a $\$5$ bill and evaluates two game options using expected profit calculations.

• Option 1: Roulette ($\$5$ Bet on Number 7)

  • Outcome 1 (Loss): Lose $\$5$ ($X = -5$). Probability P(-5) = \n\nfrac{37}{38}.

  • Outcome 2 (Win): Net gain of $\$175$ ($X = +175$; gross payout is $\$180$ minus initial $\$5$ wager). Probability P(+175) = \n\nfrac{1}{38}.

  • Expected Value Calculation:         E = \left(-5 \times \n\nfrac{37}{38}\right) + \left(175 \times \n\nfrac{1}{38}\right)         E = \n\nfrac{-185 + 175}{38} = \n\nfrac{-10}{38} \approx -0.26\,\text{dollars}

  • Units & Meaning: Units remain in dollars ($-\$0.26$). This represents an expected loss of $26\,\text{cents}$ ($26\,\text{pennies}$) per game played.

• Option 2: Craps ($\$5$ Bet)

  • Outcome 1 (Loss): Lose $\$5$ ($X = -5$). Probability P(-5) = \n\nfrac{251}{495}.

  • Outcome 2 (Win): Net profit of $\$5$ ($X = +5$). Probability P(+5) = \n\nfrac{234}{495}.

  • Expected Value Calculation:         E = \left(-5 \times \n\nfrac{251}{495}\right) + \left(5 \times \n\nfrac{234}{495}\right)         E = \n\nfrac{-1255 + 1170}{495} = \n\nfrac{-85}{495} \approx -0.07\,\text{dollars}

  • Units & Meaning: Expected loss of $7\,\text{cents}$ ($-\$0.07$) per game played.

• Comparative Decision Analysis:

  • Long-Run repeated play (100 rounds):

    • Craps is the mathematically superior choice in the long run.

    • Craps has a higher (less negative) expected value ($-\$0.07$ vs. $-\$0.26$).

    • Over $100$ games, expected cumulative loss in Craps is $\approx \$7.00$, compared to an expected loss of $\approx \$26.00$ in Roulette.

  • Single-Trial play (Only 1 wager of $\$5$):

    • Roulette may be preferred by a risk-averse player playing a single game due to the higher potential payout ($\$175$ profit vs. $\$5$ max profit in Craps).

    • Rational decision criteria shift depending on whether an action represents a single trial or long-run repeated trials (Law of Large Numbers).