Theoretical Distributions From a Model

Lesson 1 of 2: Computing Each Probability

In this lesson:

• Recognize when probabilities come from a model
• Find the probability of one specific result
• Count the arrangements to get each P(X = k)

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Recognize a theoretical distribution as one computed from a model
2. Find the probability of one specific sequence using independence
3. Build for each value using the count

Predict the Scores Before the Test Exists

A student guesses on all five questions. Each has four choices.

= number correct, values 0 to 5.

Can we predict each score's chance before anyone takes it?

The Model: One in Four, Independent

• Four equally likely choices, pure guessing →
• So on each question
• The questions are independent of one another

Theoretical Means Computed, Not Collected

• Every probability here comes from the model
• We never administer the test or tally results
• Contrast: the next standard gets probabilities from data

One Specific Sequence by Multiplication

Get question 1 right, the other four wrong:

"And" Means Multiply, Not Add

For independent events, combine by multiplying

How Many Sequences Give k Correct?

One sequence with 2 correct has probability .

But many sequences give exactly 2 correct. How many?

That count is the missing piece.

Ten Ways to Get Exactly Two Correct

There are orders.

Multiply the Shared Probability by the Count

• Each of the 10 orders has probability
• So

The General Formula for Each Value

• = number of ways to place the correct answers

Build the Whole Table Over 1024

Using a common denominator of .

Your Turn: Compute P of X Equals 3

Use :

Work it out over 1024 before advancing.

Now Solo: Compute P of X Equals 4

No scaffolding this time.

Find the one-sequence probability, then multiply by .

Report your answer over 1024.

Two Counting Traps to Avoid

Don't drop the factor — many orders give the same k

The values of X are not equally likely — different k have different counts

What You Built In This Lesson

✓ A theoretical distribution is computed from the model

✓ One specific sequence: multiply independent probabilities

✓ Each : one-sequence probability times

Coming Up Next: Grade It and Graph It

Next lesson: confirm the table sums to 1, graph its right-skewed shape, and compute the expected grade.

Then compare grading schemes — which one discourages guessing?