Predict Results from a Fair Coin

Part 1

To start, we want to calculate the probability of getting exactly 20 heads out of 30 coin flips if the coin is fair. A fair coin means there's a 50% chance of flipping heads and a 50% chance of flipping tails on any given toss.

Python
# P(X=20) X~Binomial(0.5, 30)
import scipy.stats as sc

sc.binom.pmf(20, 30, 0.5) #2.8%

Part 2

P-values help us understand if the observed results (like getting 20 heads) are extreme enough to doubt a null hypothesis (in this case, that the coin is fair).

Python
#H0: the coin is fair
#H1: the coin is not fair

# p-value is the probability of obtaining 20 heads in 30 flips or something more extreme (farther from a fair coin)
# if the null is true

# probability of 20, 21, 22, ..., 30 heads out of 30 flips

1-sc.binom.cdf(19, 30, 0.5) #4.9%

The result suggests that getting 20 or more heads is quite unlikely (less than 5% chance) if the coin is fair. It's a hint that the coin might not be fair, although there's still a small chance (4.9%) these results could occur with a fair coin.

Part 3

Now, let's simulate flipping a fair coin 30 times, repeating this experiment 100,000 times to see how often we get 20 or more heads. This simulation approach gives us another way to understand the likelihood of our observed outcome.

Python
import numpy as np

np.random.binomial(1, 0.5) # random flip of the coin
np.random.binomial(30, 0.5) # 30 random flips of the coin

results = [np.random.binomial(30, 0.5) for _ in range(100000)]
sum([1 if val == 20 else 0 for val in results])/100000

Part 4

And if we're interested in finding out how often we get 20 or more heads, not just exactly 20 (similar to how we calculate p-values):

Python
import scipy.stats as ss
ss.binom.pmf(20, 30, 0.5)

# pvalue is the probability of this results (20 heads) or one more extreme (farther from a fair coin) if
# the coin is truly fair

# probability of 20, 21, 22, ... 30
sum([1 if val >= 20 else 0 for val in results])/100000

1-ss.binom.cdf(19, 30, 0.5)

This simulation method confirms our earlier calculations, showing again that getting 20 or more heads is unusual for a fair coin, supporting the idea that our coin might not be fair. However, it also illustrates that there's a slight chance (similar to the 4.9% we calculated before) of these results happening with a fair coin.

By comparing these approaches, we reinforce our understanding of statistical analysis and hypothesis testing, providing a comprehensive view of how to interpret the results of our coin flips.