CLT: 60 Heads in 100 Flips

This probability question is about using a normal approximation for a binomial experiment in a classic quant prep setting. You move from a discrete model of repeated independent events to a continuous distribution to estimate tail probabilities efficiently. It focuses on understanding how a large collection of identical trials aggregates into a smooth, bell-shaped distribution, a core idea in probability distributions used across quant interviews.

It trains comfort with the Central Limit Theorem in a concrete, numerical way, including identifying the correct mean and variance of the sum of outcomes. It also reinforces intuition about how likely substantial deviations from the average are when the number of trials is large. Overall, it checks whether candidates can turn an abstract probabilistic limit theorem into a practical approximation tool under time pressure.

This matters in quant interviews because normal approximations underpin risk estimation, PnL aggregation, and portfolio modeling. Interviewers want to see if you can handle large-sample reasoning quickly, make sensible approximations, and assess tail risks without exact combinatorics. Mastering this kind of CLT-based reasoning is essential in quant prep, since many real-world pricing and risk questions rely on similar approximations rather than exact closed forms.

What it tests

Whenever you have a large sum of independent, identically distributed random variables, their distribution tends toward a normal distribution regardless of the original variable's distribution—this is the Central Limit Theorem (CLT). The key is that the mean and variance of the sum are simply the sum of the means and variances of the individual variables. This means that even if the original variable is discrete (like a coin flip), the sum's distribution becomes approximately continuous and bell-shaped as the number of trials grows. The CLT works because the cumulative effect of many small, independent random influences smooths out irregularities, making the normal distribution a universal approximation for sums. This is why, for large $n$, the binomial distribution can be replaced by a normal distribution with matching mean and variance for probability estimates.