60 Heads: Quant Interview Question with Central Limit Theorem in Python | QuantGuide IO

Описание: Solve the "60 Heads" problem from QuantGuide IO with this step-by-step statistics tutorial! The problem: A fair coin is tossed 100 times. Using the Central Limit Theorem (CLT) without continuity correction, estimate the probability of observing at least 60 heads, rounded to 5 decimal places. We’ll approximate the binomial distribution with a normal distribution, compute the z-score, and find the probability as 0.02872. Perfect for math enthusiasts, quant interview prep, or anyone learning statistics and probability!

🔍 *What You'll Learn:*
Understanding the 60 Heads problem setup
Applying the Central Limit Theorem to approximate a binomial distribution
Calculating the z-score and probability using a normal distribution
Using Python’s scipy.stats to compute the result

Python code to compute the probability
import scipy.stats as stats

Parameters for the binomial distribution
n = 100 # Number of tosses
p = 0.5 # Probability of heads (fair coin)
x = 60 # At least 60 heads

Mean and standard deviation
mean = n * p # 100 * 0.5 = 50
sd = (n * p * (1-p)) ** 0.5 # sqrt(100 * 0.5 * 0.5) = sqrt(25) = 5

Z-score using CLT (no continuity correction)
z = (x - mean) / sd # (60 - 50) / 5 = 2

Probability P(X {greater than}= 60) = 1 - P(Z {less than} z)
probability = round(1 - stats.norm.cdf(z), 5)
print(f"Z-score: {z}")
print(f"Probability of at least 60 heads: {probability}")
Output:
Z-score: 2.0
Probability of at least 60 heads: 0.02872

🌟 *Why Solve 60 Heads?*
This problem is a classic quant interview question using the Central Limit Theorem, perfect for quant finance interviews or math competitions! It teaches you how to approximate binomial probabilities with a normal distribution, a key skill for statistics. The probability of at least 60 heads is 0.02872, and we’ll show you exactly how to get there step by step. Master this, and you’ll be ready for more advanced statistics challenges!

📚 *Who’s This For?*
Math and statistics enthusiasts
Students prepping for quant interviews
Anyone practicing for math competitions

👍 Like, subscribe, and comment: What statistics problem should we solve next? Next up: More quant interview questions—stay tuned!

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