How many dice to roll

Optimizing the number of dice to roll for a winning outcome

This is a medium-difficulty probability question that tests your ability to set up and optimize a discrete outcome probability. Rather than computing a single probability, you must find the value of a parameter that maximizes it—a common pattern in quantitative trading and risk interviews.

The problem requires you to express the probability of success as a function of the number of dice rolled, then determine where that function peaks. You'll need to reason about how the probability of exactly one or exactly two target outcomes changes as you add more trials, and recognize when the trade-off tips in your favour. The solution involves comparing probabilities across different parameter values and identifying the maximum without necessarily computing every single case.

• Binomial probability and counting outcomes
• Discrete optimization and extrema
• How probability distributions shift with sample size