Scaffolding with AI Guided Learning

Aug 11, 2025

Knowledge acquisition scaffolding with AI

Below is a conversation between a pretend student and Gemini with Guided Learning enabled. A new concept of Probability and Statistics is introduced to the student layer by layer. Can you find the following features in this converstion?

  • The beggining of the conversation is based on intuition and some 'world' prior knowledge.

  • The concept of independence is broken down into smaller, manageable parts

    • The idea that one event does not influence another
    • Probability enters midway through the conversation
    • The mathematical notation and formulas follow later

  • Guided transition from intuition to formal language

    • The student arrives at the formal definition not because it was told to them, but because it was presented as a direct translation of their own logic.

  • Reinforcement and confirmation

    • Positive reinforcement is present throughout the conversation

The conversation

Student: My course notes say two events are "independent," but I don't really get what that means.

AI: Excellent question. That's a fundamental concept in statistics.

In simple terms, two events are independent if the outcome of one has absolutely no effect on the outcome of the other.

Imagine you flip a coin and roll a die at the same time. The coin can land on heads or tails. The die can land on any number from 1 to 6.

If the coin lands on heads, does that make any of the die numbers (1, 2, 3, 4, 5, 6) more or less likely to come up?

Student: Of course not.

AI: You've got it. The result of the coin flip doesn't change the physics of the die roll, so they are independent.

Now, let's apply that same logic to just one object.

Imagine you roll a die and get a 3. Then you pick up that same die and roll it again. Does that first roll of a 3 have any effect on what you'll get on the second roll? 🤔

Student: If this a fair die, no.

AI: That's exactly right. The die has no memory. The first roll of a 3 doesn't make a 3 (or any other number) more or less likely on the second roll. The two rolls are independent.

I like that you mentioned a "fair die." That's a very important detail.

Assuming a die is fair means that every one of the six sides has an equal chance of landing face up. This brings us to the next key idea: probability 🎲.

Probability is just a way to measure the chance of a particular event happening. We usually write it as a fraction:

Probability= Total number of possible outcomes/Number of desired outcomes

So, for a single roll of a fair die, what's the probability of rolling a 5?

Student: it is 1/6.

AI: That's exactly right. There's one "5" on the die (the desired outcome), and six possible sides in total. So the probability is 1/6.

Now, let's combine our two concepts: independence and probability.

To find the probability of two independent events both happening, we multiply their individual probabilities. This is often called the "AND rule."

P(A and B) = P(A) × P(B)

So, let's try it. What's the probability of rolling a 5 on the first roll AND a 2 on the second roll?

Student: 1/36. But how can I get dependent events from a die?

AI: That's exactly right! The probability of rolling a 5 is 1/6, the probability of rolling a 2 is 1/6, and since they are independent, you multiply them together: 1/6×1/6=1/36.

Now for your question about dependent events. That's a great way to think about it—what's the opposite of independent?

For events to be dependent, the outcome of the first event must change the possible outcomes, or the probabilities, of the second event. With a single die, this is hard because the die itself doesn't change after a roll.

So, let's change the game a little. Imagine you have a small bag containing three dice:

Two are red 🔴🔴

One is blue 🔵

You reach into the bag without looking and pull one die out. Then, without putting the first die back, you pull out a second one.

Let's think about the second draw. Does the color of the first die you pulled out affect the probability of the second die you pull out being blue?

Student: Yes it does. If I pulled the blue die and I hold it is not possible to pull a blue die again.

AI: You've hit on the key idea perfectly. Because you don't put the first die back, the outcome of the first draw directly changes the possible outcomes for the second draw. That's the very definition of dependent events.

Calculating the Probability

Let's put some numbers to your logic. We'll look at the probability of the second draw, which depends on the result of the first.

Scenario 1: Let's say you draw a red die 🔴 first and set it aside.

Now, only two dice are left in the bag. What's left, and what is the new probability of drawing the blue die 🔵?

Student: 1/2

AI: Exactly. After you take out a red die, there's one red die and one blue die left in the bag (two total). So, the probability of drawing the blue one is 1/2.

Notice how the probability changed. Initially, the chance of drawing a blue die was 1/3. But after drawing a red one first, the chance became 1/2. This change is the key feature of dependent events.

Now for the second scenario you brought up. Let's reset the bag to two red 🔴🔴 and one blue 🔵.

This time, you draw the blue die 🔵 first. What is the probability of drawing a blue die on your second draw?

Student: Nill. How would you write this with formulas?

AI: You're right, the probability is zero. You can't draw a blue die if it's already out of the bag! We can write that probability as just 0.

Now, for your question about formulas. This is a great time to introduce the notation for conditional probability. "Conditional" is just a formal word for an event that depends on another.

Conditional Probability Notation

When the probability of event A depends on event B, we write it like this:

P(A | B)

We read this as "The probability of A happening, given that B has already happened." The vertical bar "|" means "given."

So, for the scenario we just did, we would write:

P(Blue on 2nd | Blue on 1st) = 0

This is the formal way of saying, "The probability of drawing a blue die on the second try, given that we already drew a blue die on the first try, is 0."

Using this P(A | B) format, how would you write out our first scenario, where we calculated the probability of drawing a blue die after first drawing a red one?

Student: p(A|B)=1/2

AI: You're so close! The probability is indeed 1/2. For the notation, we just need to replace the general "A" and "B" with descriptions of our specific events.

Remember:

  • A is the event we want to find the probability of.

  • B is the condition we already know is true.

In that scenario, what was the event we wanted to happen (A), and what was the condition that came first (B)?

Student: B is red of first draw and A is blue on the second draw?

AI: Exactly. So, putting it all together, the formal notation would be:

P(Blue on 2nd | Red on 1st) = 1/2

This statement clearly says, "The probability of drawing a blue die on the second try, given that we drew a red die on the first try, is 1/2."

You've done a great job distinguishing between independent and dependent events. We've seen that:

  • Independent events don't affect each other (like two rolls of a die).

  • Dependent events change the probabilities of what comes next (like drawing dice from a bag without replacement).

Where would you like to go from here? We could explore:

  1. How to calculate the probability of two dependent events both happening.

  2. Describing the results of many die rolls (using concepts like mean, median, and mode).

ITI