Choose the better bet

Learning Objectives Model real-world scenarios involving rates and averages using rational functions. Calculate the expected value of a probabilistic scenario to quantify its long-run outcome. Analyze the horizontal asymptotes of rational functions to determine long-term trends and limiting values. Compare two or more rational functions by finding their intersection points and analyzing their behavior. Apply limits to rational functions to evaluate the behavior of a bet as the number of trials approaches infinity. Justify a decision between two options by interpreting the mathematical properties of their corresponding rational functions. Would you rather have a 1/3 chance to win $90 or a 1/10 chance to win $250? 🤔 Let's use rational functions to find the winning strate...

TermDefinitionExample Rational FunctionA function that can be written as the ratio of two polynomial functions, P(x) and Q(x), in the form f(x) = P(x) / Q(x), where Q(x) ≠ 0.If a batter has 30 hits in 100 at-bats and then gets x consecutive hits, their new batting average is A(x) = (30 + x) / (100 + x). Expected Value (E[X])The long-run average outcome of a random event, calculated by summing the products of each possible outcome and its corresponding probability.A coin flip pays $2 for heads (P=0.5) and $0 for tails (P=0.5). The expected value is E[X] = ($2)(0.5) + ($0)(0.5) = $1. Horizontal AsymptoteA horizontal line y = L that the graph of a function approaches as x approaches ∞ or -∞. In betting scenarios, it represents the long-term average value or success rate.For A(x) = (30 + x) /...

Expected Value Formula E[X] = \sum_{i=1}^{n} x_i P(x_i) To find the expected value of a bet, multiply each net outcome `x_i` by its probability `P(x_i)` and sum the results. A higher expected value indicates a better bet in the long run. Long-Run Average (Limit at Infinity) For f(x) = \frac{ax^n + ...}{bx^m + ...}, \lim_{x \to \infty} f(x) = \begin{cases} a/b & \text{if } n=m \\ 0 & \text{if } n<m \\ \infty \text{ or } -\infty & \text{if } n>m \end{cases} Use this to find the horizontal asymptote, which represents the limiting average value of a bet or success rate as the number of trials (x) becomes very large. Comparing Bets To find when Bet A is better than Bet B, solve the inequality A(x) > B(x). This helps determine the conditions (e.g., num...