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3.1 Introduction to Probability

Section Overview

The Idea of Probability

A big fact emerges when we watch coin tosses or the results of random sampling and random assignment closely: chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long run. After many repetitions, the Law of Large Numbers states that we can call this long-run relative frequency of a chance outcome its probability. We will explore this further in Example 3.1.1.

Example 3.1.1

Explain in terms of probability why the proportion of heads for the first 20 tosses is so variable but stabilizes to 0.5 after 500 tosses.

Example 3.1.1 — recorded answer

Say when you first start tossing a coin that the proportion of heads start at 0 on the first toss (because it landed on tails), rises to 0.5 when the second toss gives a head, then falls to 0.33, 0.25, and 0.2 as we get more tails.

After that, say the proportion of heads begins to increase, rising well to 0.5 before it decreases again.

But suppose we keep tossing the coin until we have made 500 tosses. As we have seen, the proportion of tosses that produce heads is quite variable at first. As we make more and more tosses, however, the proportion of heads gets close to 0.5 and stays there.

Interactive 3.1.1 graphically displays this example problem if this concept isn’t quite

Source gap (S0203): The recorded answer stops after “if this concept isn’t quite” in the source. Its missing ending has not been supplied.

Interactive 3.1.1 Example 3.1.1

Line graph of the proportion of heads, with toss number from 1 to 500 on the horizontal axis and proportion from 0 to 1 vertically. The line changes sharply in earlier tosses and oscillates closer to 0.5 in later tosses. Two source callouts point to Earlier tosses and Later tosses. Open full-size image S0203
Earlier tosses — recorded annotation
In the earlier tosses, the probability is changing more suddenly.
Later tosses — recorded annotation
In the later tosses, the probability begins to change more slowly and converge to 0.5

Myths about Short-Run Regularity

The idea of probability seems straightforward. It answers the question “What would happen if we did this many times?” In fact, both the behavior of random phenomena and the idea of probability are a bit subtle. In fact, our intuition about randomness tries to tell us that random phenomena should also be predictable in the short run. When they aren’t we look for some explanation other than chance variation.

Example 3.1.2

If you toss a coin 5 times, which of the following outcomes are more probable?

HTHTT TTTHH

Example 3.1.2 — recorded answer

At first glance, it would seem that HTHTT is more probable because it looks more “random”. However, both outcomes are equally likely. The coin has no memory and thus there is no “effort” put into trying to create a more random-looking sequence.

Simulation

Imitating chance behavior based on a model that precisely reflects the situation is called a simulation. Simulations are commonly carried about in the following ways:

  • Physical devices such as cards, coin, or dice
  • Technology such as a calculator’s random number generator
  • A table of random digits such as Figure 3.1.1. A table of random digits usually starts at row 101 and usually ends at row 200. For our purposes, however, we’ve only included up to row 118.

All of these methods will be explained more clearly in Example 3.1.3

Figure 3.1.1

Source: Introduction to the Practice of Statistics, 6th Edition

RowGroup 1Group 2Group 3Group 4Group 5Group 6Group 7Group 8
1011922394534057562871396409125314254482853
1027367647145994000192727754426488242536290
1034546771759775580009532863294858222690056
1045271138889930746022740011858484876752573
1059559294007699719148160779537911729759335
1066841735013155297276585089575675021147487
1078273957890208074751181676553009438314893
1086094072024178682494361790906568796418883
1093600919365154123963885453468168348541979
1103844848789183382469739364420067668808758
1118148669487605130929700412712382764939950
1125963688804046347119719352730898489845785
1136256875206403250369971080225531148611776
1144514932992757306628003819562020293875915
1156104177684943222475973698145263189332592
1161445926056314248037165103622535049061181
1173816798532621837563223417261854144875532
1187319032533044752966984407907856595686382
Example 3.1.3

An organic foods supermarket, Friendly Foods, has decided to raffle off store gift cards to shoppers during the first six weeks of its inaugural opening. There is a 410\frac{4}{10} probability that a shopper could win a gift card. Out of 100 separate repetitions, simulate a scenario of least 4 out of 5 consecutive shoppers in the checkout line winning a gift card using:

a) A physical device

b) A calculator’s random number generator

c) A table of random digits

Example 3.1.3 — recorded answer

a) Out of the physical devices suggested earlier in the section, playing cards may seem to be the best option because there are enough cards to perform the simulation. We could use the ace up to 10 to represent all of the possibilities of the raffle, with 7, 8, 9, and 10 represent all of the possibilities of winning. Each of the five shoppers could be represented by five different decks. One complication with this approach is that since the cards would not be replaced, the probability of one shopper winning would affect the chances of the next shopper. Then there are face cards… since that gets complicated, we could also opt to use a 10-sided die as shown in Gallery 3.1.1, where rolling a 7, 8, 9, or 10 for one shopper wouldn’t affect the chances of the next shopper, and rolling the dice 5 consecutive times represents one repetition. This simulation method tells us that out of 100 repetitions, 10 result in at least 4 out of 5 shoppers winning a gift card, making the probability 10100\frac{10}{100}.

b) Slide 2 in Gallery 3.1.1 shows the randInt function randomly generating integers 1 through 10. This method works similarly to the 10-sided die because generating a 4 does not prevent the function from generating a 4 in the future.

c) For the random digits table, we will use lines 101 to 112 and part of 113 for a total of 100 repetitions. Each repetition has 5 random digits for the 5 consecutive shoppers. The condition for winning is if a shopper has the number 7, 8, 9, or 0. The results are shown in Gallery 3.1.1. The highlighted sections are the repetitions where at least 4 of the shoppers won a gift card. There are 12 repetitions highlighted out of 100, making the probability of 4 out of 5 shoppers winning a gift card 12100\frac{12}{100}. This is feasible because it is very close to the probability from rolling the 10-sided die in (a), and it took far less time. Yay, random digits tables!

The displayed die outcomes, in source order:

RepetitionRoll 1Roll 2Roll 3Roll 4Roll 5
878875
1628667
30919108
41958710
49810738
58108869
63810489
761091083
89771092
100867107
Source gap (S0209): The last die row is partly clipped in the screenshot. Its visible upper digits are transcribed above. The source also labels Rep 16 a repetition with at least four winning rolls, although its displayed digits include 2, 6, and 6. This original inconsistency is retained.

The random digits displayed in Gallery 3.1.1 frame 3:

RowGroup 1Group 2Group 3Group 4Group 5Group 6Group 7Group 8
1011922395034057562871396409125314254482853
1027367647150994000192727754426488242536290
1034546771709775580009532863294858222690056
1045271138889930746022740011858484876752573
1059559294007699719148160779537911729759335
1066841735013155297276585089570675021147487
1078273957890208074751181676553009438314893
1086094072024178682494361790906568796418883
1093600919365154123963885453468188348541979
1103844848789183382469739364420067668808708
1118148669487605130929700412712382764939950
1125963688804046347119719352730898489845785
1136256870206403250369971080225531148611776
Source gap (S0211): The gallery’s digit table differs from Figure 3.1.1 in several cells. Both original versions are retained separately.
Source gap (S0211): Source-outline inconsistency: the recorded answer and caption claim that all 12 outlined groups contain at least four winning digits, using 7, 8, 9, or 0 as winners. A check of the visible outlined digits finds four groups that satisfy this rule and eight that do not. The recorded 12/100 claim is retained as source text.

Pop Quiz 3.1.1

Question 1 of 5

If a fair coin is tossed four times and the outcomes are HHHH, then:

(1) The probability of getting four heads in a row is 116\frac{1}{16}

(2) It is more likely that the coin will land on tails than heads on the next flip

(3) A more likely sequence would have been THHT or THTH

Question 2 of 5

In a deck of cards the probability of getting one King is about 0.08. This means that:

Question 3 of 5

There are 7 dogs and 3 cats at a kennel, represented by the ace through the 10 card. Which of the following sequence of cards best simulates choosing 3 cats and 1 dog?

Four card-sequence choices. Top left: ace, 3, 2, 6, all hearts. Top right: 7, 3, 8, ace, all hearts. Bottom left: ace of hearts, ace of hearts, 2 of hearts, 6 of diamonds. Bottom right: ace of hearts, ace of spades, 6 of diamonds, 2 of clubs. Choices are also available as text below. Open full-size image S0214
Question 4 of 5

If a sports team wins 35% of the time, then which way would be the best way to assign random digits?

Question 5 of 5

From question 4, using the following sequence of random digits, how many wins are there out of the first 10 games?

82734 71490 20467 47511 81676 55300 94383 14893

Source gap (S0212): All five question prompts and choices are captured. The screenshots show no selected or checked answers; the original answer feedback is unavailable.
Glossary and deeper reading · 4 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

  • Random sampling

    Random sampling uses a specified chance process to select individuals from a population. It supports population inference when the sampling frame and data collection are appropriate, but does not guarantee perfect balance or eliminate every source of bias. For example, a random-number generator can select student identifiers from an enrollment list.

    Read more: Random sampling in the captured Statistics reading →

  • Random assignment

    Random assignment uses chance to allocate experimental units to treatment conditions. It tends to balance other influences across groups and supports causal inference in a well-conducted experiment, without guaranteeing identical groups. For example, shuffled labels can assign plants to two fertilizer treatments.

    Read more: Random assignment in the captured Statistics reading →

  • Probability

    Probability quantifies how likely an event is under a specified chance model. It lies between 0 and 1, with larger values representing greater likelihood. For example, drawing a red card from a well-shuffled standard deck has probability one half.

  • Simulation

    A simulation imitates a chance process using a specified model. Repeated simulated outcomes can approximate probabilities when the model faithfully represents the situation. For example, random digits can model repeated draws from a bag with known color proportions.

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