Probability, Expected Value, and Bayes' Theorem in Real-World Contexts

Expected Value and Casino Economics

• Probability of Winning vs. Losing: In a game with $12$ identifying numbers out of a total of $38$ numbers on a wheel (typical of American roulette wheels):     - The probability of winning ($P(\text{Win})$) is $\frac{12}{38}$.     - The probability of losing ($P(L)$) is $\frac{38 - 12}{38} = \frac{26}{38}$.     - Because the probability of losing is higher than winning, the expected value will naturally be negative over the long term.

• Expected Value ($E[x]$) Calculation:     - Definition: The expected value is the sum of each outcome ($x$) multiplied by its respective probability ($P(x)$).     - Formula: $E[x] = \sum (x \times P(x))$     - In the example provided, a player bets $\$200$ on a "two-to-one" payout. This means if they win, they gain $\$400$. If they lose, they lose the $\$200$ bet.     - Calculation: $E[x] = (400 \times \frac{12}{38}) + (-200 \times \frac{26}{38})$     - Result: $-10.53$     - Interpretation: On average, for every bet of $\$200$, the player will lose $\$10.53$ in the long run ($a \text{ day, a week, a year, or 250,000 years}$). Even though wins occur during short sessions, the statistical average remains negative.

• Casino Strategies and Design:     - Casinos deliberately set payouts so that the expected value is always negative for the player.     - In Las Vegas, many casinos have added a "third green" (triple zero) to the roulette wheel to further increase the house edge (reducing player winning probability).     - European wheels often lack the second green space (double zero), sometimes mandated by law, which offers slightly better odds to the player compared to American wheels.     - Every game in a casino is designed such that the house has "the edge."

• Fair Game Calculations:     - A "fair game" is defined as a scenario where nobody wins or loses over the long term, meaning the expected value is exactly zero ($E[x] = 0$).     - To determine what the winning payout ($f$) should be for a $\$200$ bet with a $\frac{12}{38}$ chance of winning:         - $0 = (f \times \frac{12}{38}) - (200 \times \frac{26}{38})$         - $f \times \frac{12}{38} = 200 \times \frac{26}{38}$         - $f = \frac{200 \times 26}{12}$         - $f = 433.33$     - Conclusion: For the game to be fair, the payout must be $\$433.33$. Since casinos only pay $\$400$, the player is essentially "giving away" approximately $\$33.33$ every time they play.

Probability Trees and Conditional Probability

• Conceptual Framework:     - A probability tree summarizes the results of various choices and distributions.     - Multi-stage decisions: For example, an initial survey asks travelers to choose between the Grand Canyon or New York City, then asks about their preferred activity in that location.

• Mathematical Rules for Trees:     - Summative Rule: Each individual set of branches in a specific section must add up to a probability of $1$.     - Multiplication Rule: To find the probability of a specific outcome at the end of a branch (Event A and Event B), you multiply the probabilities along that specific path.     - Addition Rule: If there are multiple ways to reach a specific outcome (e.g., "sitting by the pool" in either location), you add those branch-end probabilities together.

• Example: Vacation Activities Survey (15 travelers):     - Probability of choosing the Grand Canyon (GC): $\frac{9}{15}$     - Probability of choosing New York City (NYC): $\frac{6}{15}$     - Within GC, people choose activities like sitting by the pool, visiting local attractions, or hiking. The distribution of these preferences creates conditional probabilities.     - Within NYC, the distribution differs (e.g., more people choose museums than hiking or pools).

Bayes' Theorem and Medical Testing (Cologuard Case Study)

• Introduction to Cologuard:     - Product developed at the University of Wisconsin and latter sold to Abbott in Chicago.     - Serves as a screen for colorectal cancer (a substitute for colonoscopies).

• Test Accuracy and Reliability:     - No medical test is $100\%$ accurate.     - Sensitivity (Correct Positive): If a patient has cancer, the test is correct $95\%$ of the time ($0.95$).     - False Negative: If a patient has cancer, there is a $5\%$ chance ($0.05$) the test will incorrectly return a negative result. This is dangerous because it provides a false sense of security.     - Specificity (Correct Negative): If a patient is healthy, the test is correct $94\%$ of the time ($0.94$).     - False Positive: If a patient is healthy, there is a $6\%$ chance ($0.06$) the test will incorrectly return a positive result, causing unnecessary anxiety.

• Applying Bayes' Theorem:     - This theorem allows us to calculate the probability of having the disease given a positive test result ($P(\text{Cancer} | \text{Positive})$).     - Population Prevalence: Data from the American Cancer Society indicates the lifetime risk of colorectal cancer for men is approximately $1$ out of $25$ ($P(\text{Cancer}) = \frac{1}{25} = 0.04$). Consequently, $P(\text{Healthy}) = \frac{24}{25} = 0.96$.     - Calculating the Probability of Cancer after One Positive Result:         - Numerator: $P(\text{Cancer}) \times P(\text{Positive Test} | \text{Cancer}) = \frac{1}{25} \times 0.95 = 0.038$         - Denominator (All ways to get a positive result): $(0.038) + (P(\text{Healthy}) \times P(\text{Positive Test} | \text{Healthy}))$         - Denominator Calculation: $0.038 + (\frac{24}{25} \times 0.06) = 0.038 + 0.0576 = 0.0956$         - Final Calculation: $\frac{0.038}{0.0956} \approx 0.397$     - Result: There is only a roughly $40\%$ chance the patient actually has cancer after one positive test result. There is a $60\%$ chance they are healthy.

• Impact of Multiple Tests:     - If a physician orders a second test following a positive result, the probability increases significantly.     - If the second test is also positive, the reliability increases to approximately $91.3\%$.     - This demonstrates the mathematical significance of serial testing or seeking a "second opinion"/different diagnostic (like a colonoscopy).

Questions & Discussion

• Question (Paul): "In Vegas and most casinos they've added a third green because they wanna screw you over even more."

• Response: The speaker confirms this, noting that European wheels don't have that second green (double zero) and that the laws likely enforced this to prevent the house edge from becoming too extreme, although the casinos would prefer otherwise.

• Question (Lina): "What do the probabilities add up to?"

• Response: They add up to one. Every set of branches on a probability tree must sum to $1$.

• Question (Dante): "So, on the probability tree, how come it adds up to 15 after you split up the nine?"

• Response: The speaker clarifies that the $15$ represents the total population polled. The branching represents the conditional probability—what happens given you are already in a specific category (e.g., in New York vs. the Grand Canyon). The proportions split depending on what events occurred prior.

• Question (Matthew): "Why bother spending $\$900$ or $\$800$ doing this test if only $40\%$ of the time you really have cancer?"

• Response: The speaker notes that the test is a starting point. A physician uses it to decide whether to run another test or move to a more invasive procedure like a colonoscopy. Having a $40\%$ chance is significantly higher than the initial population risk of $4\%$, making it a useful, if imperfect, screening tool.