Card 13

🧠 📊 🔄

You're 70% sure your friend is lying. They pass a lie detector that's 80% accurate. How sure should you be now?

💭 How to Think About This

Before the test, you were 70% confident they were lying. The test says they're truthful, but it's only 80% accurate. Should you now be 80% sure they're truthful? 30%? Somewhere else? This is BAYESIAN UPDATING—the mathematically correct way to combine prior beliefs with new evidence.

After the test, how confident should you be that they're lying?

🔒 Write your thinking above to unlock hints

If evidence doesn't change your beliefs at all, you're not reasoning rationally! The test provides information. Yes, it's imperfect (80% accurate), but it SHOULD shift your confidence somewhat. Completely ignoring evidence is as bad as completely ignoring your prior.

Imagine 100 friends in this situation. 70 are lying, 30 are truthful. Test results: Of 30 truthful → 24 pass, 6 fail. Of 70 lying → 14 pass (false negative!), 56 fail. Among those who PASS: 24+14=38 total, 24 truthful. So 24/38 = 63% truthful, 37% lying!

BAYESIAN UPDATING: New belief = (Prior × Likelihood) / Normalization. Your prior (70% lying) combines with the evidence (passed test) to produce a posterior (37% lying). Neither prior alone nor evidence alone—BOTH matter! Strong priors need strong evidence to overturn.

If you were 99% sure they're lying, one 80% test wouldn't make you believe them. If you were 50/50, the test would matter more. Both your starting confidence AND the evidence strength determine the final answer. That's rational belief updating!

You should now be ~37% sure they're lying—the evidence shifted your belief significantly!

Ignoring evidence entirely isn't rational. Here's how to think about it:

The calculation (100 similar cases):

• 70 lying: 14 pass (20% error), 56 fail

• 30 truthful: 24 pass (80% correct), 6 fail

• Among 38 who passed: 24 truthful, 14 lying

• 24/38 = 63% truthful, so 37% lying

The Bayesian principle:

• Prior + Evidence → Updated belief

• Neither alone determines the answer

• Strong evidence can overturn weak priors

• Weak evidence only nudges strong priors

Your habit: When you get new evidence, ask: "How much should this shift my belief?" Don't ignore it, but don't let it override everything either.

You understand Bayesian updating! The prior (70% lying) combines with the evidence (passed 80% accurate test) to produce a posterior (~37% lying). Neither the prior nor the evidence alone determines the answer—both are weighted appropriately.

Think of 100 cases: 70 lying, 30 truthful. Test: 24 truthful pass, 6 fail; 14 lying pass (errors!), 56 fail. Among 38 who passed: 24 truthful + 14 lying. So 24/38 ≈ 63% truthful, 37% lying. The math confirms your intuition!

This is how RATIONAL belief change works. Doctors combine symptoms + tests + base rates. Scientists update theories with each experiment. You should update opinions gradually with each piece of evidence. Bayesian updating is the math of learning from experience!

You can use this framework intuitively: • "I was 80% sure of X, this weak evidence nudges me to 70%" • "I was 50/50, this strong evidence moves me to 80%" • The stronger your prior, the more evidence needed to change it. Calibrate your confidence to evidence!

Correct! ~37% lying—you demonstrated excellent Bayesian reasoning!

You understood that both prior belief and new evidence contribute to the answer.

The mechanics:

• Prior: 70% lying, 30% truthful

• Evidence: Passed 80% accurate test

• Posterior: 37% lying, 63% truthful

• Evidence shifted belief by ~33 percentage points

The Bayesian principle:

• New belief = Prior weighted by evidence

• Strong priors resist weak evidence

• Weak priors yield to strong evidence

• This is rational belief updating

Applications:

• Medical diagnosis (symptoms + tests + base rates)

• Scientific inference (theory + experiment)

• Everyday reasoning (beliefs + observations)

Your skill: You can now think about how much evidence SHOULD change your mind—not too little, not too much.

You're letting the test override your prior too much! Yes, the test is 80% accurate, but you started with strong evidence they were lying (70% confident). An 80% test shouldn't flip a 70% belief to only 20%. Your prior knowledge still carries weight!

Imagine 100 friends: 70 lying, 30 truthful. Test results: 24 truthful pass, 6 fail; 14 lying pass (20% error!), 56 fail. Among 38 who passed: 24 truthful, 14 lying. So 14/38 = 37% still lying! The test helps but doesn't eliminate your prior suspicion.

If someone claims they saw a unicorn, one blurry photo (80% reliable) shouldn't convince you! Your strong prior (unicorns don't exist) resists weak evidence. Similarly, your 70% suspicion shouldn't collapse to 20% from one imperfect test. Strong priors need strong evidence to overturn.

You should be ~37% sure they're lying—not 20%. The evidence shifted your belief significantly (from 70% to 37%), but didn't override it completely. If you'd started at 50/50, the test would move you more. Your starting point affects how much you should update!

The answer is ~37% lying—not 20%. Your prior still matters!

You were too quick to let the test override your prior belief.

The calculation:

• Prior: 70% lying (strong suspicion)

• Test: 80% accurate (moderately reliable)

• Result: 37% lying (suspicion reduced, not eliminated)

Why 20% is too low:

• The test has 20% false negative rate

• Of the 70 liars, 14 would still pass!

• Your prior knowledge isn't erased by one test

The Bayesian principle:

• Strong priors resist weak evidence

• You'd need stronger evidence (multiple tests, 99% accuracy) to move to 20%

• Single pieces of evidence nudge beliefs, they don't override them

Your habit: Ask: "Is this evidence strong enough to override what I already knew?" Usually, it just updates your belief, not replaces it.

🤔 Which thinking lens(es) did you use?

Select all the lenses you used:

Learn more about the 7 Thinking Lenses →

👨‍👩‍👧 For Parents & Teachers

🌱 A Small Everyday Story

Meera was 90% sure she'd left her keys at home.
She checked her bag—not there (confirms home theory).
Then she found a receipt from a cafe she visited.
Maybe she left keys at the cafe?
New evidence updated her belief—now 60% home, 40% cafe.
She called the cafe first. Keys found!

See more guidance →

🧠 Thinking habits this builds:

Updating beliefs proportionally to evidence strength
Not ignoring prior knowledge when new evidence arrives
Not ignoring new evidence because of strong prior beliefs
Understanding that belief change should be gradual, not all-or-nothing

🌿 Behaviors you may notice (and reinforce):

"That changes my estimate, but not completely" reasoning
Considering both prior beliefs and new evidence
Quantifying belief changes: "I was 70% sure, now I'm 50%"
Asking "How strong is this evidence?" before updating

How to reinforce: When discussing beliefs and evidence, model the updating process: "I thought X, but this evidence shifts me toward Y—maybe I'm now 60/40 instead of 80/20."

🔄 When ideas are still forming:

Some learners may find the math intimidating or think it's only for experts. Help them see that the PRINCIPLE—combining prior and evidence—can be applied intuitively even without formal calculation.

Helpful response: "You don't need exact numbers. Just ask: Does this evidence push my belief a little or a lot? Should my prior still have weight?"

🔬 If you want to go deeper:

Explore Bayes' Theorem and its formula
Discuss how spam filters use Bayesian classification
Look at how medical diagnostic algorithms work

Key concepts (for adults): Bayes' theorem, prior probability, posterior probability, likelihood, evidence, belief updating, probabilistic reasoning, inference.

You're 70% sure your friend is lying. They pass a lie detector that's 80% accurate. How sure should you be now?

🔄 Compare All Perspectives

📌 "Still ~70%"

🔄 "~37% (updated)"

📊 "~20% (test overrides)"

🤔 Which thinking lens(es) did you use?