दिन 27: Data Science के लिए Probability

import numpy as np
rng = np.random.default_rng(42)
for n in [10, 100, 10_000, 1_000_000]:
    rolls = rng.integers(1, 7, n)
    print(f"{n:>9,} dice rolls → P(six) ≈ {(rolls == 6).mean():.4f}   (true = {1/6:.4f})")

import numpy as np
import matplotlib.pyplot as plt
rng = np.random.default_rng(7)
flips = rng.integers(0, 2, 3000)
running = flips.cumsum() / np.arange(1, 3001)
plt.figure(figsize=(9, 3.4))
plt.plot(running, color="#4f46e5"); plt.axhline(0.5, color="#dc2626", ls="--")
plt.xscale("log"); plt.ylim(0, 1)
plt.title("Law of large numbers: share of heads settles at 0.5"); plt.xlabel("number of flips (log scale)")
plt.show()

import pandas as pd
t = pd.read_csv("telecom_churn.csv")
A = t["senior_citizen"] == 1
B = t["contract"] == "Monthly"
pA, pB, pAB = A.mean(), B.mean(), (A & B).mean()
print(f"P(senior)={pA:.3f}  P(monthly)={pB:.3f}  P(both)={pAB:.3f}")
print(f"P(senior or monthly) by rule = {pA + pB - pAB:.3f} | counted directly = {(A | B).mean():.3f}")
print(f"P(not churn) = {1 - (t['churn'] == 'Yes').mean():.3f}")

import pandas as pd
t = pd.read_csv("telecom_churn.csv")
churn = t["churn"] == "Yes"
print(f"P(churn)               = {churn.mean():.3f}")
print(f"P(churn | Monthly)     = {churn[t['contract'] == 'Monthly'].mean():.3f}")
print(f"P(churn | Two year)    = {churn[t['contract'] == 'Two year'].mean():.3f}")
print(f"P(Monthly | churn)     = {(t.loc[churn, 'contract'] == 'Monthly').mean():.3f}   ← not the same thing!")
print(pd.crosstab(t["contract"], t["churn"], margins=True))

import pandas as pd
t = pd.read_csv("telecom_churn.csv")
churn = (t["churn"] == "Yes")
both = churn[(t["contract"] == "Monthly") & (t["tech_support"] == "No")].mean()
print(f"P(churn | monthly AND no tech support) = {both:.3f}")
print(f"Lift over average = {both / churn.mean():.2f}x")

import pandas as pd
t = pd.read_csv("telecom_churn.csv")
churn = t["churn"] == "Yes"
for col in ["gender", "contract"]:
    rates = churn.groupby(t[col]).mean().round(3).to_dict()
    print(col, rates)
print("Gender: rates ≈ equal → churn looks independent of gender. Contract: very different → dependent.")

p_fraud = 0.001          # 1 in 1,000 payments is fraud (prior)
sens = 0.99              # P(flag | fraud)
fpr = 0.02               # P(flag | normal)
p_flag = sens * p_fraud + fpr * (1 - p_fraud)
post = sens * p_fraud / p_flag
print(f"P(flag) = {p_flag:.4f}")
print(f"P(fraud | flag) = {post:.3f}  → only {post * 100:.1f}% of flagged payments are fraud")

import numpy as np
rng = np.random.default_rng(1)
n = 1_000_000
fraud = rng.random(n) < 0.001
flag = np.where(fraud, rng.random(n) < 0.99, rng.random(n) < 0.02)
print("Flagged payments:", flag.sum(), "| real fraud among them:", (flag & fraud).sum())
print(f"Simulated P(fraud | flag) = {(flag & fraud).sum() / flag.sum():.3f}")

# Should we give a ₹200 retention coupon to a customer with 40% churn risk?
p_churn = 0.40
monthly_bill, months_left = 700, 12
coupon = 200
p_churn_with_coupon = 0.28             # assumed from a past test
ev_no = (1 - p_churn) * monthly_bill * months_left
ev_yes = (1 - p_churn_with_coupon) * monthly_bill * months_left - coupon
print(f"Expected revenue without coupon: ₹{ev_no:,.0f}")
print(f"Expected revenue with coupon:    ₹{ev_yes:,.0f}  → {'give' if ev_yes > ev_no else 'skip'} the coupon")