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To solve the problem of finding the probability that a fair coin lands heads-up more than 550 times out of 1000 flips, we can use the properties of the binomial distribution and apply the normal approximation due to the large number of trials (n = 1000).
Understanding the Binomial Distribution:
The number of heads in 1000 flips of a fair coin follows a binomial distribution with parameters:
The probability mass function (PMF) of a binomial distribution is given by:
P(X=k)=(kn)pk(1−p)n−k
where (kn) is the binomial coefficient.
Normal Approximation to the Binomial Distribution:
Standardizing the Variable:
We need to find P(X>550).
Convert this to a Z-score using:
Z=σX−μ
For X = 550:
Z=15.81550−500=3.16
Using the Standard Normal Distribution:
Conclusion:
scipy.stats
module:from scipy.stats import norm
# Parameters
n = 1000
p = 0.5
# Mean and standard deviation
mu = n * p
sigma = (n * p * (1 - p))**0.5
# Calculate Z
Z = (550 - mu) / sigma
# Probability
probability = 1 - norm.cdf(Z)
print(probability) # Output: 0.0008
This method provides a quick and efficient way to calculate probabilities for large binomial distributions using normal approximation.