Find valid range of q for outcome probabilities

You have a 3-outcome experiment with mutually exclusive outcomes: - Outcome 1 occurs with probability \(q\) - Outcome 2 occurs with probability \(q^2\) - Outcome 3 occurs with probability \(1 - q - q^2\) 1) Find the full range of \(q\) for which these form a valid probability distribution. 2) Suppose there are three tradable “tickets” (assets) and you are allowed to take long/short positions. Each ticket has a known payoff in each of the 3 outcomes. Let: - \(x \in \mathbb{R}^3\) be the portfolio positions (positive = long, negative = short) - \(p \in \mathbb{R}^3\) be today’s prices of the tickets - \(A \in \mathbb{R}^{3\times 3}\) be the payoff matrix where \(A_{ij}\) is the payoff of ticket \(j\) in outcome \(i\) Write the “construct an arbitrage portfolio” condition as a matrix/vector inequality system (i.e., a feasibility problem using \(A\), \(p\), and \(x\)).