Extension Week 1: Bayesian Updating
How New Evidence Should Change Your Beliefs
In the main curriculum, we talked about making predictions and checking outcomes. But we didn't address a crucial question: When you get new information, how MUCH should it change your mind?
Bayesian updating (named after Reverend Thomas Bayes) gives us a framework: Your new belief = your old belief, shifted by how strongly the evidence points one way or the other. You don't throw away everything you knew before. And you don't ignore the new evidence. You BLEND them.
This is one of the most powerful ideas in all of science. And it's simpler than it sounds.
Use this extension if:
- The learner has completed at least Weeks 1–11 of the core curriculum (Phases 1–3).
- The learner showed strong engagement with Week 3 (Prediction & Calibration) and Week 10 (Signal vs. Noise).
- The learner is comfortable with the idea of rating their confidence in a belief/prediction.
Hold off if:
- The learner struggled with Week 3's probability concepts.
- The core 18-week curriculum hasn't been completed yet — finish the capstone first if possible.
- This extension week is more conceptual and mathematical than the main curriculum. Recommended for students who flew through Weeks 1-18 and want more.
- The gumball bag demo is essential — it makes an abstract concept tactile and fun.
- You do NOT need to teach Bayes' Theorem as a formula. The intuition is what matters.
- Target age: best for ages 9+ or advanced younger students.
Facilitator Preparation
- Prepare two opaque bags (can't see inside):
- Bag A: 7 red and 3 blue marbles/counters/candies
- Bag B: 3 red and 7 blue marbles/counters/candies
- The student doesn't know which bag is which. Label them 1 and 2 (so the student doesn't know which is "mostly red" and which is "mostly blue")
- Have paper for tracking belief updates
- Prepare the evidence strength examples for Session 2
The key insight: Reasonable people can disagree on beliefs, but they should update in the same direction when presented with the same evidence. If someone refuses to update when the evidence clearly points one way, that's a thinking error.
For Younger Learners (Ages 8–9)
Simplest version of the concept: "When you learn something new, it should change how sure you are about what you believe. New clues update your best guess."
What to shorten or skip:
- Focus on the Mystery Bag and Weather Detective activities. They're concrete and playable.
- Skip Bayes' theorem notation entirely. Keep it intuitive: "Before the clue, I thought ___. After the clue, I think ___."
- Skip formal prior/posterior language with younger learners. Use "first guess" and "updated guess."
- Keep sessions to 20 minutes.
Adapting the activities:
- For the Mystery Bag: Use a real bag with colored blocks. Start with 3 blocks (2 red, 1 blue). Pull one out without looking. "Before you looked, what color did you THINK it was? Now you see it's red — does that change your guess about what's left?"
- For Weather Detective: Use real or pretend forecasts. "The weather app says 40% rain. You look outside and see clouds. Now what do you think — more or less than 40%?"
- Keep the number of clues to 2–3 per round. Don't overwhelm.
Journal alternative: "Something I changed my mind about this week: I used to think ___. Then I learned ___. Now I think ___." Spoken or drawn is fine.
What success looks like: The learner can describe how one piece of new evidence changed their belief — even informally.
- Introduce the concept with numbers: "Before = 50%, new evidence pushes it to 75%."
- Work through 2–3 Bayesian reasoning problems with simple fractions.
- Discuss real examples where people fail to update: "Why do people keep believing something even when new evidence says otherwise?"
Guided Session 1
The Belief Update
Learning Goal
By the end of this session, the student can:
- explain that beliefs should change when new evidence arrives
- demonstrate belief updating by adjusting confidence after each piece of evidence
- explain why prior beliefs matter (you don't start from zero each time)
Activities
In Week 3, you practiced calibration — making your confidence levels accurate. Today we learn the rule for HOW to shift those levels when new evidence arrives.
1. The Gumball Bag Mystery
Setup:
"I have two bags. One has MOSTLY red gumballs. The other has MOSTLY blue. I've picked one bag, but I won't tell you which. You have to figure it out by pulling gumballs one at a time."
Before any pulls:
"Right now, what's the probability that I'm holding the mostly-red bag?"
Answer: 50% — you have no information, so both bags are equally likely. Write it down.
Pull 1: Reach in without looking and pull out one gumball. (Put it back after showing.)
Let's say it's RED.
"How does this affect your belief? If the bag were mostly red, pulling a red is likely (70%). If the bag were mostly blue, pulling a red is less likely (30%). So the evidence points toward... mostly red!"
"Update your belief: maybe now you're 70% confident it's the mostly-red bag."
Pull 2: Pull another. Let's say it's RED again.
"Two reds. The mostly-red bag explanation is getting stronger. Maybe you're now 85% confident."
Pull 3: Pull another. This one is BLUE.
"Interesting — a blue! Does this mean it's the mostly-blue bag? Not necessarily. Even the mostly-red bag has 30% blue. But it does shift your confidence back a bit. Maybe 75% now."
Continue for 5-7 pulls. Track the confidence level after each pull:
| Pull # | Color | Confidence (Mostly Red) | Confidence (Mostly Blue) |
|---|---|---|---|
| Before | - | 50% | 50% |
| 1 | Red | 70% | 30% |
| 2 | Red | 85% | 15% |
| 3 | Blue | 75% | 25% |
| ... |
Reveal the bag at the end. Discuss:
- "How quickly did you become confident?"
- "Did the blue gumball make you doubt everything, or just shift your belief a little?"
- "That's Bayesian updating: each piece of evidence shifts your belief, but it doesn't erase everything you already knew."
2. The Key Principle
Explain the core idea simply:
"Your new belief = your old belief + the new evidence, weighted by how strong the evidence is."
- If the evidence is STRONG (very unlikely if your belief is wrong), it shifts your belief a lot.
- If the evidence is WEAK (could happen either way), it barely shifts your belief.
- You NEVER go straight to 100% or 0% on a single piece of evidence.
3. Everyday Bayesian Updating
Practice with everyday examples:
"You think your friend likes pancakes (80% confident). Then you see them leave pancakes on their plate at breakfast."
Ask: "How much does this change your belief?"
- It depends! Maybe they were full. Maybe the pancakes were bad. But it should shift your confidence DOWN somewhat — maybe to 60%.
"You think it will rain today (30% confident). Then you look outside and the sky is completely dark with thick clouds."
- Strong evidence! Update to maybe 80%.
"You think your team will win (50/50). Then you learn the other team's best player is sick."
- Moderate evidence. Maybe shift to 65%.
Guided Session 2
Strong Evidence vs. Weak Evidence
Learning Goal
By the end of this session, the student can:
- distinguish between strong and weak evidence
- explain why strong evidence should shift beliefs more than weak evidence
- identify situations where people fail to update (or update too much)
Activities
1. The Evidence Strength Scale
Not all evidence is created equal. Rate these on a 1-5 scale:
| Evidence | Strength | Why |
|---|---|---|
| One person's opinion | 1-2 | People can be wrong or biased |
| A scientific study with 10,000 people | 5 | Large, systematic, controlled |
| "I saw it on social media" | 1 | Unverified, could be anything |
| You personally experienced it 3 times | 3 | Real but small sample |
| An expert in the field says so | 4 | Knowledgeable but not infallible |
| A YouTube video claims it | 1-2 | Anyone can make a video |
| You tried it 50 times and tracked data | 4-5 | Your own systematic evidence |
2. Updating Failures
Discuss the two most common mistakes:
Mistake 1: Refusing to update (stubbornness)
"Some people hold onto their beliefs NO MATTER what evidence they see. They explain away everything. 'That study doesn't count because...' 'That's just one case...' 'I still FEEL like I'm right.'"
Ask: "Is there anything that would change your mind?" If the answer is "no," that's a red flag.
Mistake 2: Updating too much (overreacting)
"Other people swing wildly with every new piece of information. They hear one story and completely change their mind. Then they hear another story and flip back."
Strong conviction with an openness to update is the sweet spot:
"Strong opinions, loosely held." Have confidence in what you believe, but stay ready to update when the evidence is genuinely strong.
3. The Belief Audit
Pick 3 beliefs the student holds. For each one:
- How confident are you? (0-100%)
- What evidence formed this belief?
- How strong is that evidence? (1-5)
- What new evidence would make you MORE confident?
- What new evidence would make you LESS confident?
- How strong would that evidence need to be to actually change your mind?
This exercise builds the habit of holding beliefs as probability estimates rather than absolute certainties.
Independent Practice
Goal
Practice conscious belief updating in daily life.
Activities
1. The Update Log
For one week, notice moments when you receive new information about something you have a belief about. Log at least 5:
| Belief | Confidence Before | New Evidence | Evidence Strength | Confidence After |
|---|---|---|---|---|
2. The Belief That Changed
Write about a belief you held strongly that changed over the course of this curriculum. What evidence caused the shift? Was it one big piece of evidence or many small ones?
Decision Journal
Choose a belief you're unsure about. Write your current confidence level and the evidence behind it. Over the coming week, actively seek out evidence that could update this belief in either direction. Report back with your updated confidence.
Reflection Questions
- Is it a sign of strength or weakness to change your mind when you see good evidence?
- What's the difference between being "open-minded" and being "wishy-washy"?
- Can you think of a time when you should have updated your belief but didn't? What stopped you?
Quick Mastery Check
After this week, check whether the learner can:
- Update a belief: "You think there's a 50-50 chance your friend is coming to your party. Then they text 'What time should I arrive?' Now what do you think?" (Looking for: "Now I'm way more confident they're coming — like 90%.")
- Explain the process: "What does it mean to 'update' your belief?" (Looking for: "You change how sure you are based on new information.")
- Show restraint: "Is one clue always enough to completely change your mind?" (Looking for: "No — sometimes one clue only changes it a little. You need strong evidence to change a lot.")
If the learner can update a belief in the right direction with new evidence, Bayesian intuition is building.
Pause and Notice
After the Mystery Bag or belief-updating exercise, ask:
"Has someone ever showed you evidence that you were wrong, but you STILL didn't want to change your mind? What made it hard?"
"Updating beliefs can feel uncomfortable — like admitting you were wrong. But it's actually a sign of strength. The smartest people in the world change their minds ALL THE TIME when they get new evidence. The question isn't whether you're ever wrong — it's whether you're brave enough to update when you see the evidence."
This week's takeaway: Changing your mind isn't flip-flopping. It's learning.
Spiral Review
- From Week 3: "This is your Probability Glasses, powered up. In Week 3 you learned to rate your confidence. Now you're learning to UPDATE that confidence when new evidence arrives."
- From Week 10: "Signal vs. noise matters here. Only update your beliefs based on SIGNAL — new evidence that's actually relevant. Noise shouldn't change your mind."
- From Week 4: "Bayesian thinking is the opposite of hindsight bias. Instead of pretending you 'knew it all along,' you track exactly how your belief changed — and why."
- From Week 17: "Your experiment was a Bayesian update. You had a prediction (prior), collected data (evidence), and drew a conclusion (posterior). You've already done this!"