Probability

# Probability

**Topic 5 · Tuesday, January 19, and Thursday, January 21**

]

---

# Where are we?

.box-inv-1.small[Summarize data]

.box-inv-1.small[Visualize data]

???

We can describe a distribution now. Probability is the bridge from describing data to saying something about people we never observed.

---

# Where are we? From sample to everyone

.box-inv-8.small[A sample is never the whole story. Probability says how much it can fool us]

???

Book's cycle, redrawn. The bottom arrow is the one we can't do yet.

---

# The question we'll keep asking

.picture-medium[![Cartoon of a large green hump labeled population, with a small purple pile on each side, labeled sample 1 and sample 2. A note reads p = 0.02. Sample 1 asks, OK, what about now? The population sighs: Yes. I'm still sure I birthed both of you.](img/horst-p-value.png)]

.box-8.small.sp-after-half[If what I think about the population is true, how weird would a sample like mine be?]

.box-inv-8.small[And how weird must it be before I decide I was wrong?]

???

Say the first question and let it sit before showing the second. The second one is the logic of hypothesis testing, which starts in Topic 7.

---

# Plan for today

.box-1.medium.sp-after-half[What is probability?]

.box-6.medium.sp-after-half[From bars to curves]

.box-3.medium.sp-after-half[The normal distribution]

???

Three blocks of about 20 minutes. Each ends with students doing something: try, predict, predict.

---

# What is probability?

---

---

# Probability is a share of outcomes

$$
p(A) = \frac{\text{outcomes that count as } A}{\text{all possible outcomes}}
$$

.box-1.small[0.25, 1 in 4 and 25% all say the same thing]

???

Proportion, probability and percentage are the same idea here. We'll move between them all quarter.

---

# This only works if we draw at random

Everyone has the same chance of being picked
]

One draw doesn't change the chances for the next
]

.box-inv-1.small[
Together these are called *independent and identically distributed*, or i.i.d. Every formula in this course leans on it
]

???

A friend-of-a-friend survey isn't random. Picking a whole classroom isn't independent: classmates share a teacher. Picking one student and putting them back keeps the chances the same on every draw.

---

# Pretend these 619 are everyone

.box-inv-1.small[They're really a sample from Delhi. For today they're everyone, so we write μ = 11.12 and σ = 3.90]

???

Greek letters, no hats, because today we know everybody. We come back to this same trick in Topic 6, when we draw many samples from these 619 students.

---

# Pick one student: what's p(12)?

.box-1.small[p(X = 12) = 76 / 619 = 0.123]

???

Ask first, then show the answer. The tallest bar is simply the most likely score.

---

# A range of scores just adds up bars

.box-1.small[p(X ≤ 12) = (2 + 1 + 4 + … + 76) / 619 = 405 / 619 = 0.654]

---

# Try it: Read the counts

<table class="deck-table">
 <thead>
  <tr>
   <th style="text-align:center;"> Score </th>
   <th style="text-align:center;"> 9 </th>
   <th style="text-align:center;"> 10 </th>
   <th style="text-align:center;"> 11 </th>
   <th style="text-align:center;"> 20 </th>
   <th style="text-align:center;"> 21 </th>
   <th style="text-align:center;"> 22 </th>
   <th style="text-align:center;"> 23 </th>
   <th style="text-align:center;"> 24 </th>
   <th style="text-align:center;"> 25 </th>
   <th style="text-align:center;"> 26 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> Students </td>
   <td style="text-align:center;"> 55 </td>
   <td style="text-align:center;"> 66 </td>
   <td style="text-align:center;"> 60 </td>
   <td style="text-align:center;"> 4 </td>
   <td style="text-align:center;"> 5 </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> 1 </td>
  </tr>
</tbody>
</table>

.box-8.small.sp-after-half[Pick one of the 619 students at random. Find each probability]

???

About 6 minutes, with calculators. Work alone, compare with a neighbor, then hands.

---

# Try it: Answer

.box-8.small[A score of 20 or more is rare here. That's the sense of "weird" we'll use all quarter]

???

Nobody scored 22 or 25, so in this population those probabilities are exactly 0.

---

# From bars to curves

---

---

# Shares make a probability distribution

.box-inv-6.small[Same shape, new axis. Each bar is now a probability, and all the bars add up to 1]

???

Scores here are whole numbers from 0 to 26, so we can list every outcome. The word for the general idea is a probability distribution. The book reserves "density" for the continuous case, which is next.

---

# Continuous scores can't be counted

.box-inv-6.small[A continuous variable has endless possible values. The chance of any one exact value is zero]

???

Think of time spent on a test. Exactly 12.000000 minutes never happens. The bars shrink toward nothing, so we stop asking about single values and ask about ranges.

---

# Density: where probability piles up

.box-inv-6.small[Height isn't a probability. It tells you how concentrated the probability is around that point]

---

# Ask about ranges: area is the answer

.box-6.small.sp-after-half[What's the chance of 12 or below?]

.box-inv-6.small[The whole area is 1, and the dark area is 405 / 619 = 0.654: the same answer as counting]

---

# Predict it: Tall or wide?

.box-8.small.sp-after-half[Draw one person at random. Which is more likely?]

.options[
.option[
**A** The person lands in region A, where the curve is tallest
]
.option[
**B** The person lands in region B, where the curve is lower
]
]

???

About 6 minutes. Hands are fine here. Ask a few students to say why before showing the answer.

---

# Predict it: Area wins, not height

.box-inv-8.small[Region B is lower everywhere, but it's so much wider that it holds more of the area]

.box-8.small[Probability is area. Height alone can't tell you]

---

# The normal distribution

---

---

# A bell curve, set by just two numbers

.pull-left[
<img src="05-probability_files/figure-html/bell-plot-1.png" alt="A symmetric bell curve. A vertical line at the peak is labeled mean, median and mode." width="100%" style="display: block; margin: auto;" />
]

.box-inv-3.small[Most common in the middle]

.box-inv-3.small[Tapers off toward both ends]

$$
Y = \frac{1}{\sqrt{2\pi\sigma^2}}\, e^{-(X - \mu)^2 / 2\sigma^2}
$$
]

.box-3.small[Only μ and σ change. You'll never compute this by hand]

???

It's a common shape for populations, which is why we start here. Look at the formula once: only two parameters, both Greek, so the curve describes a population. Everything else is a constant.

---

# Every normal curve splits the same way

.box-inv-3.small[Each mark is one standard deviation. The areas are always the same]

???

The tail sections hold everything beyond 2, so the outer labels sit a little above the thin curve. Each side adds up to 50%.

---

# The standard normal relabels the axis

.box-inv-3.small[z = (X − μ) / σ moves the mean to 0 and the SD to 1. The shape doesn't change]

???

This is Topic 4's z-score. A standard normal distribution is a normal distribution in z-score units. A normal curve doesn't have to be standardized, but once it is, the sections sit at whole numbers.

---

# Our 619 scores stay close to the curve

.box-inv-3.small[Within 1 σ: 71% of students, 68% of the curve. Within 2 σ: 94% and 95%]

???

Close, not perfect, and that's normal for real data. The curve even gives some area to scores below zero, which can't happen.

---

# Area is probability, so we can read it off

.box-3.small.sp-after-half[Draw one person from a normal population. What's the chance of each?]

???

Both come straight from the sections on the previous slide. A probability that small is what we'll later call weird.

---

# Watch it happen, one person at a time

???

This is step 1 of the blueprint from Topic 4, read a new way. Click +1 a few times, then +50 and +500. Each dot is one person drawn at random, and panel 3 shows the share of draws in each place. That's probability as a long-run share, and the shares settle on the areas. Ignore the H₀ label for now. Tool after Kristoffer Magnusson's rpsychologist.com.

---

# Predict it: Halfway out

.box-inv-8.small[
You know that p(z > 1) = 0.1587 and p(z > 2) = 0.0228. Now pick the spot halfway between. What's p(z > 1.5)?
]

.options[
.option[
**A** About 2%
]
.option[
**B** About 7%
]
.option[
**C** About 9%
]
.option[
**D** About 16%
]
]

???

About 6 minutes. Vote, turn to a neighbor, vote again, then show the answer. C is what you get by splitting the 1-to-2 section in half. Sketching the curve shows why that's a little too much.

Andy: create this poll in Poll Everywhere, then add the `.poll-bar` button from EDUC 265's 06-01 with its presenter id.

---

# Predict it: Answer

.options[
.option.wrong[
**A** About 2%
]
.option.correct[
**B** About 7%
]
.option.wrong[
**C** About 9%
]
.option.wrong[
**D** About 16%
]
]

.box-8.small[The curve keeps falling, so more of that section sits near z = 1. To get exact values for any z, we need a table]

---

# Before Thursday

.box-inv-8.medium.sp-after-half[Finish Chapter 6]

.box-inv-8.medium.sp-after-half[Read sections 7-1 and 7-2 for the second half of class]

.box-8.medium[Bring a calculator: we'll use the table]

---

# Session 2: Thursday, January 21

.box-inv-8.small.sp-after-half[Tuesday in one line: probability is area under a curve]

.box-5.medium.sp-after-half[The unit normal table]

.box-7.medium.sp-after-half[From scores to percentiles]

.box-inv-8.small[Then Topic 6 begins: sampling distributions]

???

Two blocks of about 20 minutes, each ending with something to do: try, then play. Topic 6 takes the rest of the session.

---

# The unit normal table

---

---

# The table answers one z in three ways

.box-inv-5.small[These are the three areas for z = 1. Body and tail always add up to 1]

???

The table in the book's appendix has these same columns, labeled A to D. The exams attach a copy.

---

# A piece of the table

.box-5.small.sp-after-half[Tuesday's poll: the row for 1.50, column Tail, gives p(z > 1.5) = 0.0668]

.box-inv-5.small[The full table is at the back of the book, and exams come with a copy]

---

# Sketch first, then pick the column

.pull-left[
.steps[
1. Draw the curve and shade the area you want
2. Mark your z-scores
3. Is that a body, a tail, or a piece from the mean?
4. Read the column, then add or subtract
]
]

.pull-right[
<img src="05-probability_files/figure-html/sketch-plot-1.png" alt="A normal curve with the area from z equals minus 1 to z equals 1.5 shaded and split at the mean. The left piece is 0.3413 and the right piece is 0.4332." width="100%" style="display: block; margin: auto;" />

???

Most mistakes in this chapter come from skipping the sketch. Here the area crosses the mean, so we add two "mean to z" values.

---

# Negative z? The curve is a mirror

.box-inv-5.small[The table lists only positive z. Both tails here equal 0.0668, so drop the sign and read the tail]

---

# The table also runs backwards

.pull-left[
.box-inv-5.small[Find 0.0500 in the Tail column. It falls between z = 1.64 and 1.65, and we use 1.65]
]

???

Remember 1.96. It will come back for the rest of the quarter.

---

# A spreadsheet does the same lookup

<table class="deck-table">
 <thead>
  <tr>
   <th style="text-align:left;"> You want </th>
   <th style="text-align:left;"> Formula, in Excel or Google Sheets </th>
   <th style="text-align:center;"> Result </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Body for z = 1.5 </td>
   <td style="text-align:left;"> =NORM.DIST(1.5, 0, 1, TRUE) </td>
   <td style="text-align:center;"> 0.9332 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Tail for z = 1.5 </td>
   <td style="text-align:left;"> =1 - NORM.DIST(1.5, 0, 1, TRUE) </td>
   <td style="text-align:center;"> 0.0668 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> z for a body of 0.95 </td>
   <td style="text-align:left;"> =NORM.INV(0.95, 0, 1) </td>
   <td style="text-align:center;"> 1.645 </td>
  </tr>
</tbody>
</table>

.box-inv-5.small[The 0 and the 1 are μ and σ. The body is everything below z, so a tail is 1 minus the formula]

???

No devices on exams, so the table is still the skill to practice. The spreadsheet helps with homework.

---

# Try it: Use the table

.options[
.option[
**1** Find p(z < 0.25)
]
.option[
**2** Find p(z > −0.25)
]
.option[
**3** 90% of scores fall below which z?
]
.option[
**4** The middle 60% lies between which two z-scores?
]
]

.box-8.small[Sketch first. Work alone, compare with a neighbor, then we vote]

???

About 6 minutes. Hands for the vote. Questions 1 and 2 have the same answer, and that's the point: sketch them.

---

# Try it: Answer

.box-8.small[The first two match because the curve is symmetric. For the last one, each tail holds 20%]

---

# From scores to percentiles

---

---

# Normal curves differ only in their labels

.box-inv-7.small[Convert to z and one table serves them all. But the population must be normal, or the table's areas are wrong]

.source[Mindspark: Muralidharan, Singh, and Ganimian (2019). SAT: College Board, 2025 SAT Suite Annual Report.]

???

For a heavily skewed population the table gives wrong answers, because its areas belong to the normal shape. Always ask whether normal is reasonable.

---

# Three stops: score, z, probability

X = μ + zσ
]
.stop[z-score]
.link[
Unit normal table
]
.stop[Probability]
]

Go right: find z, then read the area
]

Go left: read z from the table, then find X
]

.box-inv-7.small[You can't jump straight from X to a probability. Every trip goes through z]

---

# How many students scored above 1200?

.box-inv-7.small[z = (1200 − 1029) / 235 = 0.73]

.box-inv-7.small[Tail for z = 0.73: 0.2327, so about 23%]
]

.pull-right[
<img src="05-probability_files/figure-html/sat-plot-1.png" alt="A normal curve for SAT totals with the area above 1200 shaded. That is z equals 0.73, and the shaded tail is about 23%." width="100%" style="display: block; margin: auto;" />
]

.box-7.small[Real count at 1200 or above: 507,341 of 2,004,965, or 25%. Close, not exact]

???

Work this one on the board and ask the room for each step. SAT totals run from 400 to 1600, so the real distribution is cut off at both ends and isn't perfectly normal.

---

# Percentile rank: the share at or below

.box-7.medium.sp-after-half[1200 has a percentile rank of 77]

.box-inv-7.small.sp-after-half[
23% scored above it, so 100 − 23 = 77% scored at or below it
]

.box-inv-7.small.sp-after-half[
It's the body of the curve at that score
]

.box-inv-7.small[
Percentile rank describes a person's place in the group. It isn't the percent they got right
]

---

# A percentile: the score at a given share

.box-inv-7.small[The 95th percentile has a tail of 0.05, so z = 1.65]

.box-inv-7.small[X = μ + zσ = 1029 + 1.65 × 235]

.box-7.small[X = 1417, or about 1420 on the SAT's scale]
]

.pull-right[
<img src="05-probability_files/figure-html/sat-95-1.png" alt="A normal curve for SAT totals with the top 5 percent shaded, beyond a score of about 1417." width="100%" style="display: block; margin: auto;" />
]

???

Percentile rank starts with a score and ends with a percent. A percentile starts with a percent and ends with a score. Scores come in steps of 10, which is why 1416 becomes about 1420.

---

# Play it: Warm-up for Friday

.box-inv-8.small.sp-after-half[Curves and percentiles. Answer as a question. Nothing is scored]

The percentile rank of z = 0
]

The share of a normal population above z = −1
]

???

About 6 minutes for all three clues. Tell a neighbor your answer to each one, then reveal.

---

# Play it: For 300

.box-inv-8.small[
Mindspark's curve has μ = 11.12 and σ = 3.90. Treating it as normal, this score is the 90th percentile. In the table, the tail 0.10 is closest at z = 1.28
]

.options[
.option[
**A** What is 6.1?
]
.option[
**B** What is 12.0?
]
.option[
**C** What is 14.6?
]
.option[
**D** What is 16.1?
]
]

???

Andy: create this poll in Poll Everywhere, then add the `.poll-bar` button from EDUC 265's 06-01 with its presenter id.

---

# Play it: Answer

.options[
.option.wrong[
**A** What is 6.1?
]
.option.wrong[
**B** What is 12.0?
]
.option.wrong[
**C** What is 14.6?
]
.option.correct[
**D** What is 16.1?
]
]

.box-inv-8.small.sp-after-half[X = 11.12 + 1.28 × 3.90 = 16.1. The real 90th percentile of our 619 students is 16]

.box-8.small[A takes the wrong sign. B skips σ, C uses 0.90 as if it were a z-score, and 0.90 is a share]

---

# From one score to a whole sample

.box-1.medium.sp-after-half[So far: how weird is one score?]

.box-3.medium.sp-after-half[Next: how weird is one sample's mean?]

.box-inv-8.small.sp-after-half[Our 619 students stay a population. We draw many samples, and their means pile up into a curve of their own]

.box-inv-8.small[Areas under that curve drive the four steps of a hypothesis test]

???

The simulator's blueprint from Tuesday was the n = 1 case. Topic 6 raises n, right after this slide.

---

# What you can do now

.box-inv-8.small[Find a probability as a share of outcomes, or as an area]

.box-inv-8.small[Say why height on a density curve isn't a probability]

.box-inv-8.small[Use the unit normal table, forward and backward]

.box-inv-8.small[Go from a score to a z-score to a probability, and back]

.box-inv-8.small[Find a percentile rank and a percentile]

---

# Next steps

Topic 6 starts: sampling distributions, from sections 7-1 and 7-2
]

Quiz, covering everything through z-scores and probability. Bring a calculator
]

.box-3.small[Mastery and Problem Set for Chapter 6 are due Sunday, January 24 at 6 PM]

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