How Much Would You Stake on a Bet with Positive Expected Value?

The recruiter describes a gamble with a positive expected value and asks how much you would be willing to bet on it. The report gives no exact terms and describes the question as an observation of how you behave with risk, so treat it as both a calculation and a judgment call: commit to an amount and explain why. For practice, use this illustrative version (the numbers are an assumption, not the reported terms): a biased coin lands heads with probability 0.6. You choose a stake; on heads you win an amount equal to your stake, and on tails you lose your stake. You have \$1,000 available. ### Clarifying Questions - Is this a single bet, or can I bet repeatedly? - Is the \$1,000 all the money I have, or a bankroll set aside for this game? - Is the probability of heads known exactly, or is it an estimate? - Is there a maximum stake, and can I lose more than I put in? ### Part 1 — One bet You get exactly one bet. How much do you stake, and why? ```hint Expectation is not the whole story Write the expected gain and the spread of outcomes as functions of the stake, and ask what each outcome would mean for you relative to everything you own. ``` #### What This Part Should Cover - Expected value and standard deviation of the result as functions of the stake - How the answer depends on the size of the bankroll relative to your total wealth, and on your attitude to risk - A committed amount with a reason ### Part 2 — Repeated bets Now you can bet again and again on independent flips, each time staking a fixed fraction of your current money. What fraction do you choose, and why not more or less? ```hint Think in growth rates Over many bets, wealth multiplies rather than adds. Consider which quantity a fixed fraction should maximize when outcomes compound. ``` #### What This Part Should Cover - Why staking everything on every flip leads to ruin despite having the highest expected value - The fraction that maximizes long-run growth, and how it follows from the probability and the payout - How uncertainty about the probability changes the fraction you would actually use ### What a Strong Answer Covers - Clear expected value and variance arithmetic - Sizing relative to the bankroll rather than as a fixed amount, with a stated attitude to risk - Growth-optimal sizing for repeated bets, and the danger of betting too much - An adjustment for an edge that is only estimated - A decisive answer, explained calmly and consistent between the two parts ### Follow-up Questions - The payout becomes 2 to 1 on heads, and the probability of heads becomes 0.4. Do you still bet, and how much? - You learn that the probability of heads is somewhere between 0.55 and 0.65. What do you change? - Would your answer change if the \$1,000 were borrowed? - How would you size the bets if you were managing someone else's money?