Week 5 · Chapter

Chapter 5 — Probability Foundations

Introduction to Statistics Free sample edition

The module's primary reading · ~35–45 minutes · Objective 4 (apply the basic rules of probability, including conditional probability and independence, and compute and interpret the expected value and standard deviation of a discrete random variable, including binomial settings) · Read it first this week; the lecture, the tutorial, the quiz, and the Data Lab all build on it.

In plain language

Chance is part of almost every decision you make, from carrying an umbrella to trusting a test result. This week you learn to put a number on chance, called a probability (a number from 0 to 1 that says how often something happens in the long run). You will list every way a situation can turn out and count the ways you care about. You will learn three short rules for "not", "or", and "and". Then you will see how one piece of news, such as which island a penguin lives on, can change a chance, which is the idea of a conditional probability (a chance worked out after you learn something new).

Why you already care about this

A weather app says there is a 30% chance of rain tomorrow. A board game turns on whether two dice add up to 7 on your next roll. A friend has lost a coin toss 5 times in a row and says the sixth toss is "due" to go their way. Each of these is a claim about chance, and each one can be read well or badly.

You already reason about chance without the vocabulary. You leave early when traffic is likely to be heavy. You buy the larger bag when the small one might run out. What you may not have yet is a clear way to check your gut. Is the friend right about the coin? Does knowing one fact about a penguin change your best guess about its species? This chapter gives you the counting habits and the rules that answer questions like these.

By the end of this chapter you will be able to:

explain what a probability means as a long-run share, and say why a streak does not make an outcome "due";
list the 36 ways two dice can land and find the chance of an event by counting;
use the complement, addition, and multiplication rules on dice, the calendar, and a guessed quiz;
read a conditional probability off the penguin species-by-island table, and show that the order of the condition matters;
check whether two events are independent, and explain why mutually exclusive events are not independent.

Carry this sentence with you: a chance is a long-run share — not a promise about the next try.

Where we are

Already covered: Weeks 1 through 4 taught you to describe data you already have: where the data came from, what they look like, their center and spread, and how two variables move together.
This week: probability, the language of chance: long-run frequency, sample spaces and events, the complement, addition, and multiplication rules, conditional probability, and independence.
How they connect: a relative frequency from a table becomes a probability when you imagine the process repeated many times, and Week 4's conditional distribution becomes a conditional probability.
Where we're going next: this week opens a new unit on chance; Week 6, Random Variables, will attach numbers to outcomes and find the average a chance process produces in the long run.

Each earlier week feeds this one. Week 1 taught you the difference between a sample and a population, and the penguin field log is a sample of 344 birds. Week 2 taught you to turn a count into a relative frequency, a count divided by the total, and to read a two-way table of species by island. Week 3 gave you the mean and the spread of a list. Week 4 taught you to read a conditional distribution, the share of each category inside one row or one column of a two-way table. You will reuse that last move in Section 6, where it becomes a conditional probability.

1. Randomness: the long run settles, the short run swings

Hold a fair six-sided die in your hand. Before you roll it, you cannot say which face will come up. After 600 rolls, though, you can say a lot. About one roll in six will show a six. That mix of "unpredictable once" and "predictable over many tries" is the heart of this week.

Randomness is the property of a process whose single outcomes cannot be predicted in advance but whose outcomes follow a stable pattern over many repetitions, like the roll of a fair die.
A long-run relative frequency is the share of a very large number of repetitions of a random process in which an event happens; it is the everyday meaning of a probability.
A probability is a number from 0 to 1 that gives the long-run share of repetitions in which an event happens; 0 means it never happens and 1 means it always happens.

We write the probability of an event in a short form: P(six) means "the probability of rolling a six". You read it out loud as "P of six".

Worked example — sixes on a fair die. A fair die has 6 faces, and each one is equally likely to land on top.

Count the faces that give a six. Only 1 face shows a six.
Divide by the number of faces. P(six) = 1 ⁄ 6, about 16.7%.
Picture a long run. In 600 rolls you would expect about 600 × 1 ⁄ 6 = 100 sixes.

In words: over many rolls, about one roll in 6 shows a six. That is a statement about the long run. It says nothing certain about the next roll.

Watch what happens to the share of sixes as the rolls pile up. After 6 rolls, you could easily see no sixes at all, which is a share of 0%, or two sixes, which is a share of about 33.3%. Both are far from 16.7%. After 600 rolls, the share almost always sits near 16.7%, even though the count of sixes may land a few above or below 100. The share settles; the short run swings.

The law of large numbers is the fact that, as a random process is repeated more and more times, the share of repetitions in which an event happens settles toward that event's probability, even though short runs can swing widely.

Limitation: the law of large numbers is a promise about shares over many tries. It does not promise that the count of sixes ever lands on exactly 100, and it makes no promise at all about any single roll.

Misconception check — the streak that is "due". Students often think that after five rolls with no six, a six is "due" on the next roll. It feels fair, as if the die owes you one. But the die has no memory. On the sixth roll, the chance of a six is still 1 ⁄ 6. The long run evens out because the next 600 rolls swamp the streak, not because the die makes up for it. This belief is called the gambler's fallacy: the mistaken belief that an outcome becomes more likely after a streak without it; independent trials such as die rolls have no memory, so the chance stays the same. Your friend with the coin toss has fallen for it too.

In practice (your tools). You can watch the law of large numbers in a spreadsheet. In Google Sheets, type =RANDBETWEEN(1,6) in cell A2, then copy it down to A601 to get 600 rolls. In an empty cell, type =COUNTIF(A2:A601,6)/600 to get the share of sixes, and format it as a percent with one decimal. Every edit makes the sheet roll again, so you get a new share each time. Now try =COUNTIF(A2:A7,6)/6 for the first six rolls only, and compare how much more that short-run share jumps around. In Excel, the same two formulas work, and pressing F9 rolls again.

Memory hook for dice and coins alike: short runs swing; long runs settle.

2. Sample spaces and events: list every way it can land

Picture a board game where you roll two dice and move the total. Before you can say how likely a total of 7 is, you need a full list of what can happen. Counting starts with that list.

An outcome is one possible result of a single repetition of a random process, such as the ordered pair (3, 4) for two dice.
The sample space is the complete list of every possible outcome of a random process, such as the 36 ordered outcomes of rolling two dice.
An event is a set of outcomes from the sample space that you name and ask about, such as all the ways two dice can add to 7.
Equally likely describes outcomes that each have the same chance of happening; when all outcomes are equally likely, an event's probability is its number of outcomes divided by the total number of outcomes.

We call the sample space S (the list of every possible outcome). We name an event with a capital letter, so A (the first event we ask about) might be "the sum is 7".

Worked example — the 36 ways two dice can land. Think of one die as red and one as blue, so (2, 5) and (5, 2) are different outcomes.

Count the sample space. The red die has 6 faces and the blue die has 6 faces, so there are 6 × 6 = 36 equally likely ordered outcomes, from (1, 1) to (6, 6).
List the event "sum of 7". The outcomes (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) make 6 outcomes.
Divide. P(sum = 7) = 6 ⁄ 36 = 1 ⁄ 6.
List the event "sum of 10 or more". The outcomes (4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6) make 6 outcomes.
Divide. P(sum ≥ 10) = 6 ⁄ 36 = 1 ⁄ 6.

In words: a total of 7 comes up on about one roll in 6, and so does a total of 10 or more. If you are the player who needs a big total, you face the same odds as the player who needs a 7.

Here is the whole sample space as a grid of sums.

Sum of two fair dice for each of the 36 ordered outcomes, with the red die in rows and the blue die in columns
Red die	Blue 1	Blue 2	Blue 3	Blue 4	Blue 5	Blue 6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

Read a cell by its row and column: the row 3, column 4 cell holds 7, the sum for the outcome (3, 4). The 7s run along one diagonal, from the bottom left to the top right, and there are six of them.

Misconception check — counting sums instead of outcomes. Students often reason that the sums run from 2 to 12, which is 11 possible totals, so each total has a chance of 1 ⁄ 11. That treats the totals as equally likely, and they are not. A total of 2 needs (1, 1), one outcome. A total of 7 can happen six ways. The cure: count equally likely outcomes, the 36 ordered pairs, and only then group them into totals.

Memory hook for any game of chance: list the outcomes first — then count the ones you want.

3. The complement rule: count what does not happen

Go back to the weather app. If the chance of rain is 30%, you already know the chance of no rain: 70%. You did not need a second forecast. Rain and no rain cover every possibility, so their chances add to 100%.

The complement of an event is the event that it does not happen; it contains every outcome in the sample space that the event leaves out.
The complement rule says that the probability an event does not happen equals 1 minus the probability that it does happen.

In symbols, the complement rule reads P(not A) = 1 − P(A). It is often the quickest path, because counting what does not happen can be easier than counting what does.

Worked example — a total of 9 or less. You want P(sum ≤ 9) for two dice. Counting every outcome from 2 to 9 is slow. The complement of "sum is 9 or less" is "sum is 10 or more", and you already counted that event.

Name the complement. "Sum ≤ 9" fails exactly when the sum is 10 or more.
Use its probability. P(sum ≥ 10) = 6 ⁄ 36.
Subtract from 1. P(sum ≤ 9) = 1 − 6 ⁄ 36 = 30 ⁄ 36 = 5 ⁄ 6, about 83.3%.

In words: about five rolls in 6 give a total of 9 or less. A player who needs 10 or more should expect to wait.

Worked example — the calendar. Pick one month of a common year at random, each month equally likely. Seven of the 12 months have 31 days.

Count the event. P(31 days) = 7 ⁄ 12.
Use the complement. P(not 31 days) = 1 − 7 ⁄ 12 = 5 ⁄ 12.

In words: about 5 picks in 12 land on a month shorter than 31 days, that is, February or one of the four 30-day months. Check it by counting: 1 + 4 = 5 months.

Misconception check — a complement with a gap or an overlap. Students often write the complement of "sum is 10 or more" as "sum is 10 or less". But a sum of exactly 10 sits in both events, so they overlap. The true complement is "sum is 9 or less". The cure: an event and its complement never share an outcome, and together they cover the whole sample space. Check both conditions before you subtract.

4. The addition rule: add the chances, then subtract the overlap

Suppose a game pays you if the dice total 7 or if they total 10 or more. You win on either one. In probability, "A or B" means at least one of them happens, and that includes both.

Two events are mutually exclusive when they share no outcomes, so they cannot both happen on the same trial; they are also called disjoint.
The addition rule gives the chance that at least one of two mutually exclusive events happens: add their two probabilities.

Worked example — two events with no overlap. A total of 7 and a total of 10 or more cannot both happen on one roll, so the two events are mutually exclusive.

Write each probability. P(sum = 7) = 6 ⁄ 36, and P(sum ≥ 10) = 6 ⁄ 36.
Add. P(sum = 7 or sum ≥ 10) = 6 ⁄ 36 + 6 ⁄ 36 = 12 ⁄ 36 = 1 ⁄ 3.

In words: you win on about one roll in 3. That is twice the chance of either event alone, because the two events never share an outcome.

Many pairs of events do overlap, though. For those you need the general form.

The general addition rule works for any two events: add the two probabilities, then subtract the probability that both happen, so outcomes in the overlap are counted only once.

In symbols: P(A or B) = P(A) + P(B) − P(A and B). When the events are mutually exclusive, P(A and B) is 0, and the general rule turns into the addition rule.

Worked example — two events that overlap. Let A (the first event) be "the first die shows 6", and let B (the second event) be "the sum is 10 or more".

Count A. The first die shows 6 in 6 outcomes: (6, 1) through (6, 6).
Count B. The sum is 10 or more in 6 outcomes, listed in Section 2.
Find the overlap. The outcomes (6, 4), (6, 5), and (6, 6) are in both events, so P(A and B) = 3 ⁄ 36.
Add, then subtract. P(first die 6 or sum ≥ 10) = 6 ⁄ 36 + 6 ⁄ 36 − 3 ⁄ 36 = 9 ⁄ 36 = 0.25.

In words: a quarter of all rolls, one roll in 4, gives a first die of 6, a big total, or both. You can check by listing the 9 outcomes: (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6), (4, 6), (5, 5), and (5, 6).

Misconception check — counting the overlap twice. Students often add 6 ⁄ 36 + 6 ⁄ 36 and stop at 12 ⁄ 36. That counts (6, 4), (6, 5), and (6, 6) twice, once for each event. The cure: before you add, ask whether the two events can happen together. If they can, find the overlap and subtract it once.

Memory hook for "or": add the two — then take the overlap back out once.

5. The multiplication rule: multiply when one event leaves the other alone

Roll a red die and a blue die at the same moment. What the red die shows has no effect on the blue die. That "no effect" is what makes the next rule work.

Independent events are two events for which learning that one happened leaves the probability of the other unchanged.
The multiplication rule says that the probability two independent events both happen equals the product of their separate probabilities.

Worked example — two sixes. The two dice are independent, and each shows a six with chance 1 ⁄ 6.

Write each chance. P(red six) = 1 ⁄ 6, and P(blue six) = 1 ⁄ 6.
Multiply. P(both dice show 6) = 1 ⁄ 6 × 1 ⁄ 6 = 1 ⁄ 36.
Check against the grid. Only one of the 36 outcomes, (6, 6), has two sixes, and 1 ⁄ 36 matches it.

In words: double sixes come up on about one roll in 36.

Worked example — guessing on a quiz. A student guesses at random on a four-option multiple-choice question. One option of the four is right, so a guess is right with chance 1 ⁄ 4 = 0.25. Now the student guesses on two questions, and one guess has no effect on the other.

Write each chance. Each guess is right with chance 1 ⁄ 4.
Multiply. P(both guesses right) = 1 ⁄ 4 × 1 ⁄ 4 = 1 ⁄ 16, or 6.25%.

In words: about one student in 16 who guesses both questions gets both right. Guessing is a poor plan, and it gets worse with every added question.

Misconception check — multiplying chances that are linked. Students often multiply any two chances to get "both". That works only when the events are independent. Take a penguin from the field log at random. Every one of the 124 Gentoo penguins was logged on Biscoe, so P(Gentoo and Biscoe) is the full 124 ⁄ 344, the same as P(Gentoo). Multiplying 124 ⁄ 344 by P(Biscoe) = 168 ⁄ 344, which is less than one half, would give less than half the true answer. The cure: before you multiply, ask whether knowing one event changes the chance of the other. Section 7 shows you how to check.

Pause and reflect — choose one. Pick either of the 2 prompts; they count the same.

(a) Think of two things in your own week that you would treat as independent, such as rain and a bus running late; do you believe they really leave each other alone, and why?
(b) Think of a test, a game, or an app where you answer several questions in a row; how does the multiplication rule explain why guessing every answer correctly gets so unlikely so fast?

6. Conditional probability: shrink the world to what you know

Imagine a field biologist on the island of Biscoe who sees a penguin far down the beach. Which species is it? Without any news, the biologist might use the share of each species across all three islands. But the biologist does have news: the bird is on Biscoe. That news shrinks the world to Biscoe's penguins only.

Here is the species-by-island table you met in Week 2, with its totals.

Palmer penguins field log: number of penguins of each species on each island, with totals
Species	Biscoe	Dream	Torgersen	Total
Adelie	44	56	52	152
Chinstrap	0	68	0	68
Gentoo	124	0	0	124
Total	168	124	52	344

Check the totals before you use them. Biscoe holds 44 Adelie + 124 Gentoo = 168 penguins. Dream holds 56 Adelie + 68 Chinstrap = 124. Torgersen holds 52 Adelie. And 168 + 124 + 52 = 344, so n (the number of penguins logged) is 344.

A conditional probability is the probability that one event happens once you know that a second event has happened; keep only the outcomes in the second event and find the share of them that are also in the first.

We write a conditional probability with a vertical bar, read "given". So P(the penguin is Gentoo | the penguin is on Biscoe) means "the chance a penguin is Gentoo, given that it is on Biscoe". For short, we write P(Gentoo | Biscoe).

Worked example — Gentoo, given Biscoe. You want the chance a Biscoe penguin is Gentoo.

Shrink the world. Keep only the Biscoe column: 168 penguins.
Count the event inside it. Of those 168, there are 124 Gentoo.
Divide. P(Gentoo | Biscoe) = 124 ⁄ 168 ≈ 0.7381, rounded to four decimals at the end.
Compare with no news. Across all 344 penguins, P(Gentoo) = 124 ⁄ 344 ≈ 0.3605.

In words: about three in every four Biscoe penguins are Gentoo, while only about one in three penguins overall is Gentoo. That is 124 ⁄ 168 versus 124 ⁄ 344, a 2-fold jump, from the single piece of news "this bird is on Biscoe". A biologist landing on Biscoe should expect mostly Gentoo.

The same answer comes from a formula. Divide the chance of both events by the chance of the one you know:

P(Gentoo and Biscoe) ⁄ P(Biscoe) = (124 ⁄ 344) ⁄ (168 ⁄ 344) = 124 ⁄ 168.

The two 344s cancel, which is why shrinking the world to the Biscoe column gives the same result.

Misconception check — swapping the condition. Students often think P(A | B) and P(B | A) are the same number. Try it. P(Biscoe | Gentoo) asks a different question: of the 124 Gentoo penguins, how many are on Biscoe? All of them, so P(Biscoe | Gentoo) = 124 ⁄ 124 = 1. But P(Gentoo | Biscoe) = 124 ⁄ 168, well below 1, because Biscoe also holds 44 Adelie. The cure: say the condition out loud as "given", and divide by the total of the group you are given.

In practice (your tools). Load the field log with the =IMPORTDATA(...) formula from Week 1, so species is in column A and island in column B. In an empty cell, =COUNTIFS(A2:A345,"Gentoo",B2:B345,"Biscoe") returns 124, the count in both events. =COUNTIF(B2:B345,"Biscoe") returns 168, the count you are given. Divide the first by the second, then use Format, then Number, and set four decimals to see the conditional probability. In Excel, =COUNTIFS and =COUNTIF work the same way.

Memory hook for "given": divide by the group you were given — not by everyone.

7. Independence: check it, and do not confuse it with mutually exclusive

Two events are independent when news about one leaves the chance of the other alone. Section 5 used that idea to multiply. Now you test it. The test compares a conditional probability with a plain one. If the news changes the chance, the events are linked.

Dependent events are two events for which learning that one happened changes the probability of the other.

Worked example — two dice events, one independent pair and one dependent pair. You learn one fact about the first die and watch what it does to two chances.

Condition on the first die. Suppose you know the first die shows 6. The sample space shrinks to the 6 outcomes (6, 1) through (6, 6).
Test "sum of 7". Only (6, 1) of those six outcomes adds to 7, so P(sum = 7 | first die 6) = 1 ⁄ 6. That equals P(sum = 7) = 1 ⁄ 6. The news changed nothing, so the two events are independent.
Test "sum of 10 or more". Three of the six outcomes, (6, 4), (6, 5), and (6, 6), reach 10, so P(sum ≥ 10 | first die 6) = 3 ⁄ 6 = 1 ⁄ 2. That is not P(sum ≥ 10) = 1 ⁄ 6, so these are dependent events.

In words: a first die of 6 tells you nothing about whether the total is 7, but it triples your chance of a big total, from one roll in 6 to one roll in 2.

The penguins face the same test, and they fail it. P(Gentoo | Biscoe) ≈ 0.7381 is far from P(Gentoo) ≈ 0.3605, so learning the island changes the chance of Gentoo. Species and island are not independent.

Honesty note: with real data, a conditional share and an overall share are rarely equal to the last digit, even when two variables have no real link. A small gap can come from chance alone. Here the gap is large, and later in the course you will learn a test that says how large a gap must be before it counts as evidence.

Misconception check — independent is not the same as mutually exclusive. Students often think "independent" means "the two events cannot happen together". It means nearly the opposite. Take Adelie and Gentoo. No penguin is both, so P(Adelie and Gentoo) = 0, and the events are mutually exclusive. Now give yourself the news "this penguin is Adelie". The chance it is Gentoo drops to P(Gentoo | Adelie) = 0 ⁄ 152 = 0, which is not P(Gentoo) = 124 ⁄ 344. The news changed the chance, so the events are dependent. Now flip it: "the first die shows 6" and "the second die shows 6" are independent, yet both happen at (6, 6). The cure: mutually exclusive events with any chance of happening are always dependent, because one happening rules the other out.

Pause and reflect — choose one. Pick either of the 2 prompts; they count the same.

(a) Think of two traits you could record for students on a campus, such as commuting and working a job. Would you expect them to be independent, and what share would you compare to check?
(b) Think of two outcomes that cannot happen together in your daily life, such as being early and being late to the same class; explain in your own words why such a pair cannot be independent.

Key terms

randomness — The property of a process whose single outcomes cannot be predicted in advance but whose outcomes follow a stable pattern over many repetitions, like the roll of a fair die. (first used: 1. Randomness: the long run settles, the short run swings)
long-run relative frequency — The share of a very large number of repetitions of a random process in which an event happens; it is the everyday meaning of a probability. (first used: 1. Randomness: the long run settles, the short run swings)
probability — A number from 0 to 1 that gives the long-run share of repetitions in which an event happens; 0 means it never happens and 1 means it always happens. (first used: 1. Randomness: the long run settles, the short run swings)
law of large numbers — The fact that, as a random process is repeated more and more times, the share of repetitions in which an event happens settles toward that event's probability, even though short runs can swing widely. (first used: 1. Randomness: the long run settles, the short run swings)
gambler's fallacy — The mistaken belief that an outcome becomes more likely after a streak without it; independent trials such as die rolls have no memory, so the chance stays the same. (first used: 1. Randomness: the long run settles, the short run swings)
outcome — One possible result of a single repetition of a random process, such as the ordered pair (3, 4) for two dice. (first used: 2. Sample spaces and events: list every way it can land)
sample space — The complete list of every possible outcome of a random process, such as the 36 ordered outcomes of rolling two dice. (first used: 2. Sample spaces and events: list every way it can land)
event — A set of outcomes from the sample space that you name and ask about, such as all the ways two dice can add to 7. (first used: 2. Sample spaces and events: list every way it can land)
equally likely — Describes outcomes that each have the same chance of happening; when all outcomes are equally likely, an event's probability is its number of outcomes divided by the total number of outcomes. (first used: 2. Sample spaces and events: list every way it can land)
complement — The event that a given event does not happen; it contains every outcome in the sample space that the given event leaves out. (first used: 3. The complement rule: count what does not happen)
complement rule — The rule that the probability an event does not happen equals 1 minus the probability that it does happen. (first used: 3. The complement rule: count what does not happen)
mutually exclusive — Describes two events that share no outcomes, so they cannot both happen on the same trial; also called disjoint. (first used: 4. The addition rule: add the chances, then subtract the overlap)
addition rule — The rule for the chance that at least one of two mutually exclusive events happens: add their two probabilities. (first used: 4. The addition rule: add the chances, then subtract the overlap)
general addition rule — The form of the addition rule that works for any two events: add the two probabilities, then subtract the probability that both happen, so outcomes in the overlap are counted only once. (first used: 4. The addition rule: add the chances, then subtract the overlap)
independent events — Two events for which learning that one happened leaves the probability of the other unchanged. (first used: 5. The multiplication rule: multiply when one event leaves the other alone)
multiplication rule — The rule that the probability two independent events both happen equals the product of their separate probabilities. (first used: 5. The multiplication rule: multiply when one event leaves the other alone)
conditional probability — The probability that one event happens once you know that a second event has happened; keep only the outcomes in the second event and find the share of them that are also in the first. (first used: 6. Conditional probability: shrink the world to what you know)
dependent events — Two events for which learning that one happened changes the probability of the other. (first used: 7. Independence: check it, and do not confuse it with mutually exclusive)

Summary

A probability is a long-run share between 0 and 1, not a promise about the next try.
Short runs swing and long runs settle; a streak never makes an outcome due.
When outcomes are equally likely, a probability is the count of outcomes in the event divided by the count in the sample space.
The chance something does not happen is 1 minus the chance it does, whether the question is tomorrow's weather or a dice total.
For "or", add the two chances and subtract the overlap once; with no overlap, there is nothing to subtract.
For "and", multiply the two chances only when the events are independent.
A conditional probability divides by the group you were given, and swapping the condition usually changes the answer.
Two events are independent when news about one leaves the chance of the other unchanged; mutually exclusive events are dependent.

Check your understanding

Work these before looking at the answers — they are ungraded, and the answers (with reasoning) are at the bottom of the page.

A board game's spinner has 8 equal sections numbered 1 through 8. What is the chance the spinner lands on a number greater than 5? Use the complement rule to find the chance it lands on 5 or less.
A café records 80 orders by drink and cup size: 30 small coffees, 20 large coffees, 18 small teas, and 12 large teas. Find P(large | coffee) and P(large). Are drink and size independent for these orders?
A traveler books two connecting flights. Suppose the first is on time with chance 0.9 and the second with chance 0.8, and suppose the two are independent. What is the chance both are on time? What is the chance at least one is late?
A trivia app picks each question's category at random from five categories, each equally likely. Science has not come up in eight questions, and a classmate says science is now "due". What is the chance the next question is science, and what would you tell the classmate?

Answers (worked reasoning — read after attempting)

The 8 sections are equally likely. The numbers greater than 5 are 6, 7, and 8, which is 3 outcomes, so the chance is 3 ⁄ 8 = 37.5%. The complement of "greater than 5" is "5 or less", so its chance is 1 − 3 ⁄ 8 = 5 ⁄ 8 = 62.5%. Check by counting: 1, 2, 3, 4, and 5 make 5 outcomes, and 5 ⁄ 8 matches.
First find the totals. Coffee: 30 + 20 = 50 orders. Large: 20 + 12 = 32 orders. All orders: 50 + 30 = 80. Now shrink the world to coffee: P(large | coffee) = 20 ⁄ 50 = 0.4. With no news, P(large) = 32 ⁄ 80 = 0.4. The two are equal, so knowing the drink is coffee does not change the chance of a large cup. For these 80 orders, drink and size are independent.
The flights are independent, so multiply: 0.9 × 0.8 = 0.72. "At least one is late" is the complement of "both are on time", so its chance is 1 − 0.72 = 0.28. In words: about 28 trips in 100 would have at least one late flight. Notice how the complement saved you from listing three separate cases.
Each question's category is chosen independently, so the chance is still 1 ⁄ 5 = 20%. The app has no memory of the last eight questions. The classmate has fallen for the gambler's fallacy: a streak without science does not make science more likely. Over a long run of questions, science will make up about one question in 5, because the long run swamps the streak.

Next up in this module: the Readings & Resources page, the lecture tutorial, the practice set, and the quiz — and next week a probability distribution gives every outcome a number and a long-run average.

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