Priya and Arjun have been dating for 6 months. One day, Priya notices something suspicious – Arjun has started coming home late, hiding his phone, and smiling a lot at his screen. Priya’s friend whispers, “Girl, 90% chance he’s cheating!” 😱
Priya panics. She remembers that in their city, only 5% of boyfriends actually cheat (thankfully rare!).
So if cheating is so rare, why is her friend so sure?
This is exactly where Bayes’ Theorem comes to save the relationship (and your statistics exam).
Let’s turn Priya into a detective using Bayes’ Theorem.
Define the events:
C → Arjun is cheating (bad news)
¬C → Arjun is NOT cheating (good news)
S → Suspicious behaviour (late nights, hiding phone, secret smiles)
We know from city gossip and surveys:
P(C) = Probability a random boyfriend is cheating = 5% = 0.05
So P(¬C) = 1 – 0.05 = 0.95
P(S|C) = If he IS cheating, probability of suspicious behaviour = 90% = 0.90
(cheaters are usually bad at hiding it)
P(S|¬C) = If he is NOT cheating, probability of suspicious behaviour anyway = 20% = 0.20
(maybe he’s planning a surprise birthday party, or just addicted to Candy Crush)
Now Priya sees the suspicious behaviour (S happened).
She wants to know:
Given that he’s acting suspicious, what’s the actual probability he’s cheating?
That is: P(C|S) = ?
Bayes’ Theorem to the Rescue!
The magic formula:
P(C|S) = [P(S|C) × P(C)] / P(S)
Where P(S) = total probability of seeing suspicious behaviour (whether cheating or not).
P(S) = P(S|C)×P(C) P(S|¬C)×P(¬C)
= (0.90 × 0.05) (0.20 × 0.95)
= 0.045 0.190
= 0.235 (23.5%)
Now plug into Bayes:
P(C|S) = (0.90 × 0.05) / 0.235
= 0.045 / 0.235
≈ 0.191 or 19.1%
Even though the suspicious signs are strong (90% of cheaters show them), and even though Priya saw those signs…
there’s only a ~19% chance Arjun is actually cheating!
Priya takes a deep breath, confronts Arjun calmly…and finds out he was secretly learning guitar to serenade her on their 6-month anniversary. ❤️
#machinelearningtutorial