Probability & Expected Value - Exam Prep Notes (Comprehensive)

Key Concepts

• Random variable X
  • A function that assigns a real number to each outcome of a random process.
  • Discrete random variable has a finite or countable set of values.
• Probability distribution
  • A table (or function) listing the possible values of a random variable and their probabilities.
  • For a discrete RV, the probabilities must sum to 1: \sum_x P(x)=1.
  • Probability function P(x) assigns a probability to each value x.
  • Visuals: histogram or bar graph where height = probability P(x) and width represents the value; there are no gaps between bars for a proper discrete distribution.
• Expected Value (EV)
  • The long-run average (arithmetic mean) of the values of a random variable, weighted by their probabilities.
  • Formula: E(X)=\sumi xi P(x_i).
  • Interpreted as the average outcome if the experiment could be repeated many times.
• Examples of EV calculations (typical problems)
  • Raffle, lotteries, insurance, concerts, card games, etc. Use the EV to decide whether participation is favorable.

Probability Distribution and Expected Value (Foundational)

• Discrete distribution example: number of heads when flipping 2 coins
  • Possible values: x=0,1,2
  • Probabilities: P(0)=\frac{1}{4},\; P(1)=\frac{1}{2},\; P(2)=\frac{1}{4}
  • Expected value: E(X)=0\cdot\frac{1}{4}+1\cdot\frac{1}{2}+2\cdot\frac{1}{4}=1.
• General probability function (illustrative):
  • Example distribution: P(x) = {(0,\tfrac{1}{4}), (1,\tfrac{1}{2}), (2,\tfrac{1}{4})}.

Examples: Expected Value Calculations

• Example 1: raffle with $3 ticket price and $1,000 prize, 500 tickets sold

  • Let X be net payoff per ticket.
  • If you win: net payoff = $1000 - $3 = $997.
  • If you lose: net payoff = -$3.
  • Probabilities: P(\text{win})=\tfrac{1}{500},\; P(\text{lose})=\tfrac{499}{500}.
  • EV: E(X)=(997)\left(\tfrac{1}{500}\right)+(-3)\left(\tfrac{499}{500}\right) = \frac{997-1497}{500} = -1.
  • Meaning: you can expect to lose about $1 per ticket on average.

• Example 2: raffle for a $1000 television, 2000 tickets at $1 each

  • EV per ticket: E(X)=1000\left(\tfrac{1}{2000}\right)-1\left(1-\tfrac{1}{2000}\right)=\frac{1000}{2000}-\frac{1999}{2000}=-\frac{999}{2000}=-0.4995.
  • About a 50-cent loss per ticket on average.

• Example 3: life insurance policy

  • Policy: payout $100,000 upon death; premium = $2,600; probability of death in 10 years = 0.025.
  • Customer EV: E_{ ext{customer}} = -2600 + 100000(0.025) = -2600 + 2500 = -100.
  • Company EV: E_{ ext{company}} = 2600 - 100000(0.025) = 2600 - 2500 = 100.
  • Interpretation: customer expects to lose $100 over the 10-year period on average; company expects a $100 gain.

• Example 4: outdoor vs indoor concert decision under weather uncertainty

  • Indoors profit (certain): $17{,}000.
  • Outdoors: profits depend on rain. No rain: $74{,}000; with rain: -$157{,}000.
  • Probability of rain: 0.21; no rain: 0.79.
  • EV outdoors: E( ext{outdoors})=0.79(74000)+0.21(-157000)=58460-32970=25490.
  • Since $25,490 > 17,000$, outdoor is favorable in EV terms (assuming other constraints).

• Example 5: card-draw game from a standard deck

  • Payouts: $40 for a heart, $50 for an ace, $90 for ace of hearts; cost to draw = $15.
  • Probabilities in a 52-card deck:
  • Heart non-ace: 12 cards → probability (\frac{12}{52})
  • Ace non-heart: 3 cards → probability (\frac{3}{52})
  • Ace of hearts: 1 card → probability (\frac{1}{52})
  • Expected payoff: E( ext{payoff}) = \frac{12}{52}(40) + \frac{3}{52}(50) + \frac{1}{52}(90) = \frac{480+150+90}{52} = \frac{720}{52} \approx 13.846.
  • Net EV: E = 13.846 - 15 \approx -1.154.
  • Verdict: on average, you lose about $1.15 per draw; not favorable to play.

• Example 6: high school raffle problem

  • 3,500 tickets at $25 each; grand prize $38,000 SUV.
  • Probability to win: (1/3500).
  • EV per ticket: E = 38000\left(\frac{1}{3500}\right) - 25 = \frac{38000}{3500} - 25 \approx 10.857 - 25 = -14.143.
  • Conclusion: not a good bet; expected loss of about $14.14 per ticket.

• Example 7: horse race bet on Chebyshev (bet = $100)

  • Payouts: $575 if Chebyshev wins, $250 if 2nd, $175 if 3rd; 8 horses; equally likely to be in any final position.
  • Probability for each of top 3 finishes: 1/8 each; remaining 5 positions yield no payout.
  • Expected winnings from top-3 finishes: \frac{575+250+175}{8} = \frac{1000}{8} = 125.
  • Net EV: E = 125 - 100 = 25.
  • Verdict: positive EV; this bet is a wise move in EV terms.

• Example 8: prize structure with a big trip and multiple smaller trips

  • Grand prize: a full paid trip to Italy valued at $12{,}000.
  • Five 3rd-place prizes: trips to Las Vegas valued at $2{,}500 each (total value $12{,}500).
  • If a ticket costs t and there are N equally likely tickets, the EV per ticket is:
  • E = -t + \frac{12000}{N} + \frac{5\cdot 2500}{N} = -t + \frac{12000+12500}{N} = -t + \frac{24500}{N}.
  • Interpretation: EV per ticket depends on the ticket price t and total number of tickets N. If 24500/N > t, the ticket has positive EV.
  • Note: Without N and ticket price, you cannot compute a numeric EV; use the formula above once those values are known.

Basic Probability, Sets, and Common Rules

• Sample space and basic events
  • Sample Space S contains all possible outcomes of an experiment (e.g., coin flip, die roll).
  • Simple events: outcomes like heads, tails, or specific die face.
• Sets and Venn diagrams
  • Sets A, B, C, etc. with operations:
  • Union: A ∪ B (outcomes in A or B or both)
  • Intersection: A ∩ B (outcomes in both A and B)
  • Complement: A' (all outcomes not in A)
  • Subsets and complements can be used to compute combined probabilities.
• Probability rules (discrete)
  • If A and B are events, P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
  • Conditional probability: P(A | B) = P(A ∩ B) / P(B) provided P(B) > 0.
  • Independence (special case): A and B are independent if P(A ∩ B) = P(A)P(B).
• Odds vs. probability (conceptual)
  • Odds in favor of A is P(A) / P(A') and can be converted to probability by P = Odds / (1+Odds).
• Complements
  • P(A') = 1 − P(A).
• Example problems (illustrative patterns)
  • Example problem mixing colors or outcomes in a deck, applying P(A), P(B), and P(A ∪ B).

Sample Problems and Practice Patterns

• Sample space construction and basic set problems
  • Example: Adopting 2 kids with a gender constraint [illustrative], X = number of girls, etc. (outline shows how to build distributions from basic set counts).
• Basic distributions and expected value practice
  • Build a discrete distribution by listing outcomes and probabilities, then compute E(X).

Set Theory and Probability Practice (Additional Concepts)

• Examples show how to compute probabilities from set notation and Venn diagrams, including:
  • P(A) and P(B)
  • P(A ∪ B)
  • P(A ∩ B)
  • P(A | B) and related conditional probabilities
• Complements and independence appear in several problem patterns.

Basic Economics: Cost, Revenue, and Break-even Analysis

• Cost function
  • C(x) = m x + b, where m is variable cost per item and b is fixed cost.
• Revenue function
  • R(x) = p x, where p is price per unit.
• Profit function
  • P(x) = R(x) − C(x) = (p − m) x − b.
• Break-even point
  • Set revenue equal to cost: R(x) = C(x) → px = mx + b.
  • Solve for x: x = \frac{b}{p - m}.
  • If p > m, there is a positive break-even quantity; if p ≤ m, break-even may be impossible or require different price/cost structure.
• Practical notes
  • In some contexts, you may need to round the break-even quantity up to the next whole item (you can’t produce a fraction of a unit).
• Quick rule of thumb
  • Ensure price per unit exceeds variable cost per unit to realize any positive profit per unit and reach break-even in finite quantity.

Summary of How to Apply EV in Decision Making

• For any gamble or lottery:
  • Identify ticket price (cost), potential prizes, and probabilities of each prize.
  • Compute EV per ticket: E = \sumj (\text{payout}j) \cdot P_j - \text{cost}.
  • Positive EV suggests a favorable bet under the given assumptions; negative EV suggests it is unfavorable on average.
• When multiple outcomes exist (e.g., indoors vs outdoors, rain vs no rain):
  • Compute each scenario's EV and compare.
  • Consider external factors beyond EV (non-monetary costs/benefits, risk tolerance).

Quick Reference Formulas (LaTeX)

• Expected value of a discrete RV: E(X)=\sumi xi P(x_i).
• Probability distribution sum: \sum_x P(x)=1.
• Card-draw expected payoff (example):
  • With categories: E( ext{payoff})=\sumk xk P(xk), where xk are payoffs corresponding to outcomes.
• Indoors vs outdoors EV example: E_{ ext{outdoors}} = P( ext{no rain})\cdot(\text{profit no rain}) + P( ext{rain})\cdot(\text{profit with rain}).
• Break-even: solve px = mx + b \Rightarrow x = \frac{b}{p-m}.
• Italy/Las Vegas trip EV formula (general):
  • If there are N tickets and ticket price t, with one grand prize value $12{,}000 and five prizes $2{,}500:
  • E = -t + \frac{12000}{N} + \frac{5\cdot 2500}{N} = -t + \frac{24500}{N}.