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Contemporary Mathematics

7.10 The Binomial Distribution

Contemporary Mathematics7.10 The Binomial Distribution

Table of contents
  1. Preface
  2. 1 Sets
    1. Introduction
    2. 1.1 Basic Set Concepts
    3. 1.2 Subsets
    4. 1.3 Understanding Venn Diagrams
    5. 1.4 Set Operations with Two Sets
    6. 1.5 Set Operations with Three Sets
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  3. 2 Logic
    1. Introduction
    2. 2.1 Statements and Quantifiers
    3. 2.2 Compound Statements
    4. 2.3 Constructing Truth Tables
    5. 2.4 Truth Tables for the Conditional and Biconditional
    6. 2.5 Equivalent Statements
    7. 2.6 De Morgan’s Laws
    8. 2.7 Logical Arguments
    9. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  4. 3 Real Number Systems and Number Theory
    1. Introduction
    2. 3.1 Prime and Composite Numbers
    3. 3.2 The Integers
    4. 3.3 Order of Operations
    5. 3.4 Rational Numbers
    6. 3.5 Irrational Numbers
    7. 3.6 Real Numbers
    8. 3.7 Clock Arithmetic
    9. 3.8 Exponents
    10. 3.9 Scientific Notation
    11. 3.10 Arithmetic Sequences
    12. 3.11 Geometric Sequences
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  5. 4 Number Representation and Calculation
    1. Introduction
    2. 4.1 Hindu-Arabic Positional System
    3. 4.2 Early Numeration Systems
    4. 4.3 Converting with Base Systems
    5. 4.4 Addition and Subtraction in Base Systems
    6. 4.5 Multiplication and Division in Base Systems
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  6. 5 Algebra
    1. Introduction
    2. 5.1 Algebraic Expressions
    3. 5.2 Linear Equations in One Variable with Applications
    4. 5.3 Linear Inequalities in One Variable with Applications
    5. 5.4 Ratios and Proportions
    6. 5.5 Graphing Linear Equations and Inequalities
    7. 5.6 Quadratic Equations with Two Variables with Applications
    8. 5.7 Functions
    9. 5.8 Graphing Functions
    10. 5.9 Systems of Linear Equations in Two Variables
    11. 5.10 Systems of Linear Inequalities in Two Variables
    12. 5.11 Linear Programming
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  7. 6 Money Management
    1. Introduction
    2. 6.1 Understanding Percent
    3. 6.2 Discounts, Markups, and Sales Tax
    4. 6.3 Simple Interest
    5. 6.4 Compound Interest
    6. 6.5 Making a Personal Budget
    7. 6.6 Methods of Savings
    8. 6.7 Investments
    9. 6.8 The Basics of Loans
    10. 6.9 Understanding Student Loans
    11. 6.10 Credit Cards
    12. 6.11 Buying or Leasing a Car
    13. 6.12 Renting and Homeownership
    14. 6.13 Income Tax
    15. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  8. 7 Probability
    1. Introduction
    2. 7.1 The Multiplication Rule for Counting
    3. 7.2 Permutations
    4. 7.3 Combinations
    5. 7.4 Tree Diagrams, Tables, and Outcomes
    6. 7.5 Basic Concepts of Probability
    7. 7.6 Probability with Permutations and Combinations
    8. 7.7 What Are the Odds?
    9. 7.8 The Addition Rule for Probability
    10. 7.9 Conditional Probability and the Multiplication Rule
    11. 7.10 The Binomial Distribution
    12. 7.11 Expected Value
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  9. 8 Statistics
    1. Introduction
    2. 8.1 Gathering and Organizing Data
    3. 8.2 Visualizing Data
    4. 8.3 Mean, Median and Mode
    5. 8.4 Range and Standard Deviation
    6. 8.5 Percentiles
    7. 8.6 The Normal Distribution
    8. 8.7 Applications of the Normal Distribution
    9. 8.8 Scatter Plots, Correlation, and Regression Lines
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  10. 9 Metric Measurement
    1. Introduction
    2. 9.1 The Metric System
    3. 9.2 Measuring Area
    4. 9.3 Measuring Volume
    5. 9.4 Measuring Weight
    6. 9.5 Measuring Temperature
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  11. 10 Geometry
    1. Introduction
    2. 10.1 Points, Lines, and Planes
    3. 10.2 Angles
    4. 10.3 Triangles
    5. 10.4 Polygons, Perimeter, and Circumference
    6. 10.5 Tessellations
    7. 10.6 Area
    8. 10.7 Volume and Surface Area
    9. 10.8 Right Triangle Trigonometry
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  12. 11 Voting and Apportionment
    1. Introduction
    2. 11.1 Voting Methods
    3. 11.2 Fairness in Voting Methods
    4. 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
    5. 11.4 Apportionment Methods
    6. 11.5 Fairness in Apportionment Methods
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  13. 12 Graph Theory
    1. Introduction
    2. 12.1 Graph Basics
    3. 12.2 Graph Structures
    4. 12.3 Comparing Graphs
    5. 12.4 Navigating Graphs
    6. 12.5 Euler Circuits
    7. 12.6 Euler Trails
    8. 12.7 Hamilton Cycles
    9. 12.8 Hamilton Paths
    10. 12.9 Traveling Salesperson Problem
    11. 12.10 Trees
    12. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  14. 13 Math and...
    1. Introduction
    2. 13.1 Math and Art
    3. 13.2 Math and the Environment
    4. 13.3 Math and Medicine
    5. 13.4 Math and Music
    6. 13.5 Math and Sports
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  15. A | Co-Req Appendix: Integer Powers of 10
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index
A baseball diamond is shown with a game in progress. The pitcher is pitching the ball to a batter at home plate.
Figure 7.45 If one baseball team has a 65% chance of beating another in any single game, what’s the likelihood that they win a best-of-seven series? (credit: “baseball game” by Britt Reints/Flickr, CC BY 2.0)

Learning Objectives

After completing this section, you should be able to:

  1. Identify binomial experiments.
  2. Use the binomial distribution to analyze binomial experiments.

It’s time for the World Series, which determines the champion for this season in Major League Baseball. The scrappy Los Angeles Angels are facing the powerhouse Cincinnati Reds. Computer models put the chances of the Reds winning any single game against the Angels at about 65%. The World Series, as its name implies, isn’t just one game, though: it’s what’s known as a “best-of-seven” contest: the teams play each other repeatedly until one team wins 4 games (which could take up to 7 games total, if each team wins three of the first 6 matchups). If the Reds truly have a 65% chance of winning a single game, then the probability that they win the series should be greater than 65%. Exactly how much bigger?

If you have the patience for it, you could use a tree diagram like we used in Example 7.33 to trace out all of the possible outcomes, find all the related probabilities, and add up the ones that result in the Reds winning the series. Such a tree diagram would have 27=12827=128 final nodes, though, so the calculations would be very tedious. Fortunately, we have tools at our disposal that allow us to find these probabilities very quickly. This section will introduce those tools and explain their use.

Binomial Experiments

The tools of this section apply to multistage experiments that satisfy some pretty specific criteria. Before we move on to the analysis, we need to introduce and explain those criteria so that we can recognize experiments that fall into this category. Experiments that satisfy each of these criteria are called binomial experiments. A binomial experiment is an experiment with a fixed number of repeated independent binomial trials, where each trial has the same probability of success.

Repeated Binomial Trials

The first criterion involves the structure of the stages. Each stage of the experiment should be a replication of every other stage; we call these replications trials. An example of this is flipping a coin 10 times; each of the ten flips is a trial, and they all occur under the same conditions as every other. Further, each trial must have only two possible outcomes. These two outcomes are typically labeled “success” and “failure,” even if there is not a positive or negative connotation associated with those outcomes. Experiments with more than two outcomes in their sample spaces are sometimes reconsidered in a way that forces just two outcomes; all we need to do is completely divide the sample space into two parts that we can label “success” and “failure.” For example, your grade on an exam might be recorded as A, B, C, D, or F, but we could instead think of the grades A, B, C, and D as “success” and a grade of F as “failure.” Trials with only two outcomes are called binomial trials (the word binomial derives from Latin and Greek roots that mean “two parts”).

Independent Trials

The next criterion that we’ll be looking for is independence of trials. Back in Tree Diagrams, Tables, and Outcomes, we said that two stages of an experiment are independent if the outcome of one stage doesn’t affect the other stage. Independence is necessary for the experiments we want to analyze in this section.

Fixed Number of Trials

Next, we require that the number of trials in the experiment be decided before the experiment begins. For example, we might say “flip a coin 10 times.” The number of trials there is fixed at 10. However, if we say “flip a coin until you get 5 heads,” then the number of trials could be as low as 5, but theoretically it could be 50 or a 100 (or more)! We can’t apply the tools from this section in cases where the number of trials is indeterminate.

Constant Probability

The next criterion needed for binomial experiments is related to the independence of the trials. We must make sure that the probability of success in each trial is the same as the probability of success in every other trial.

Example 7.34

Identifying Binomial Experiments

Decide whether each of the following is a binomial experiment. For those that aren’t, identify which criterion or criteria are violated.

  1. You roll a standard 6-sided die 10 times and write down the number that appears each time.
  2. You roll a standard 6-sided die 10 times and write down whether the die shows a 6 or not.
  3. You roll a standard 6-sided die until you get a 6.
  4. You roll a standard 6-sided die 10 times. On the first roll, we define “success” as rolling a 4 or greater. After the first roll, we define “success” as rolling a number greater than the result of the previous roll.

Your Turn 7.34

Decide whether the following experiments are binomial experiments:
1.
Draw a card from a well-shuffled deck, note its suit, and replace it. Repeat this process 5 times.
2.
Draw 5 cards from a well-shuffled deck and count the number of .
3.
Draw a card from a well-shuffled deck, note whether it is a or not, and replace it. Repeat this process 5 times.
4.
Draw cards from a well-shuffled deck until you get a .

The Binomial Formula

If we flip a coin 100 times, you might expect the number of heads to be around 50, but you probably wouldn’t be surprised if the actual number of heads was 47 or 52. What is the probability that the number of heads is exactly 50? Or falls between 45 and 55? It seems unlikely that we would get more than 70 heads. Exactly how unlikely is that?

Each of these questions is a question about the number of successes in a binomial experiment (flip a coin 100 times, “success” is flipping heads). We could theoretically use the techniques we’ve seen in earlier sections to answer each of these, but the number of calculations we’d have to do is astronomical; just building the tree diagram that represents this situation is more than we could complete in a lifetime; it would have 21001.3×103021001.3×1030 final nodes! To put that number in perspective, if we could draw 1,000 dots every second, and we started at the moment of the Big Bang, we’d currently be about 0.00000003% of the way to drawing out those final nodes. Luckily, there’s a shortcut called the Binomial Formula that allows us to get around doing all those calculations!

FORMULA

Binomial Formula: Suppose we have a binomial experiment with nn trials and the probability of success in each trial is pp. Then:

P(number of successes isa)=Cna×pa×(1p)naP(number of successes isa)=Cna×pa×(1p)na.

We can use this formula to answer one of our questions about 100 coin flips. What is the probability of flipping exactly 50 heads? In this case, n=100n=100, p=12p=12, and a=50a=50, so P(flip 50 heads)=C10050×(12)50×(112)10050P(flip 50 heads)=C10050×(12)50×(112)10050. Unfortunately, many calculators will balk at this calculation; that first factor (100C50100C50) is an enormous number, and the other two factors are very close to zero. Even if your calculator can handle numbers that large or small, the arithmetic can create serious errors in rounding off.

Tech Check

Luckily, spreadsheet programs have alternate methods for doing this calculation. In Google Sheets, we can use the BINOMDIST function to do this calculation for us. Open up a new sheet, click in any empty cell, and type “=BINOMDIST(50,100,0.5,FALSE)” followed by the Enter key. The cell will display the probability we seek; it’s about 8%. Let’s break down the syntax of that function in Google Sheets: enter “=BINOMDIST(aa, nn, pp, FALSE)” to find the probability of aa successes in nn trials with probability of success pp.

Example 7.35

Using the Binomial Formula

  1. Find the probability of rolling a standard 6-sided die 4 times and getting exactly one 6 without using technology.
  2. Find the probability of rolling a standard 6-sided die 60 times and getting exactly ten 6s using technology.
  3. Find the probability of rolling a standard 6-sided die 60 times and getting exactly eight 6s using technology.

Your Turn 7.35

Compute the probabilities (rounded to 3 decimal places) of the following events related to rolling a standard 4-sided die (with faces labeled 1, 2, 3, and 4):
1.
You roll the die 10 times and get exactly four 2s.
2.
You roll the die 20 times and get exactly four 2s.
3.
You roll the die 30 times and get exactly four 2s.

The Binomial Distribution

If we are interested in the probability of more than just a single outcome in a binomial experiment, it’s helpful to think of the Binomial Formula as a function, whose input is the number of successes and whose output is the probability of observing that many successes. Generally, for a small number of trials, we’ll give that function in table form, with a complete list of the possible outcomes in one column and the probability in the other.

For example, suppose Kristen is practicing her basketball free throws. Assume Kristen always makes 82% of those shots. If she attempts 5 free throws, then the Binomial Formula gives us these probabilities:

Shots Made Probability
0 0.000189
1 0.004304
2 0.0392144
3 0.1786432
4 0.4069096
5 0.3707398

A table that lists all possible outcomes of an experiment along with the probabilities of those outcomes is an example of a probability density function (PDF). A PDF may also be a formula that you can use to find the probability of any outcome of an experiment.

Checkpoint

Because they refer to the same thing, some sources will refer to the Binomial Formula as the Binomial PDF.

If we want to know the probability of a range of outcomes, we could add up the corresponding probabilities. Going back to Kristen’s free throws, we can find the probability that she makes 3 or fewer of her 5 attempts by adding up the probabilities associated with the corresponding outcomes (in this case: 0, 1, 2, or 3):

P(makes 3 or fewer)=P(a=0)+P(a=1)+P(a=2)+P(a=3)=0.000189+0.004304+0.0392144+0.1786432=0.2223506 P(makes 3 or fewer)=P(a=0)+P(a=1)+P(a=2)+P(a=3)=0.000189+0.004304+0.0392144+0.1786432=0.2223506

The probability that the outcome of an experiment is less than or equal to a given number is called a cumulative probability. A table of the cumulative probabilities of all possible outcomes of an experiment is an example of a cumulative distribution function (CDF). A CDF may also be a formula that you can use to find those cumulative probabilities.

Checkpoint

Cumulative probabilities are always associated with events that are defined using . If other inequalities are used to define the event, we must restate the definition so that it uses the correct inequality.

Here are the PDF and CDF for Kristen’s free throws:

Shots Made Probability Cumulative
0 0.000189 0.000189
1 0.004304 0.004493
2 0.0392144 0.0437073
3 0.1786432 0.2223506
4 0.4069096 0.6292602
5 0.3707398 1

Tech Check

Google Sheets can also compute cumulative probabilities for us; all we need to do is change the “FALSE” in the formulas we used before to "TRUE."

Example 7.36

Using the Binomial CDF

Suppose we are about to flip a fair coin 50 times. Let HH represent the number of heads that result from those flips. Use technology to find the following:

  1. P(H22)P(H22)
  2. P(H<26)P(H<26)
  3. P(H>28)P(H>28)
  4. P(H20)P(H20)
  5. P(20<H<25)P(20<H<25)

Your Turn 7.36

You are about to roll a standard 6-sided die 20 times. Let S denote a success, which will be rolling a number greater than 4. Find the probabilities of the following events, rounded to 4 decimal places:
1.
S 5
2.
S < 8
3.
S > 10
4.
S 7
5.
5 < S < 8

Finally, we can answer the question posed at the beginning of this section. Remember that the Reds are facing the Angels in the World Series, which is won by the team who is first to win 4 games. The Reds have a 65% chance to win any game against the Angels. So, what is the probability that the Reds win the World Series? At first glance, this is not a binomial experiment: The number of games played is not fixed, since the series ends as soon as one team wins 4 games. However, we can extend this situation to a binomial experiment: Let’s assume that 7 games are always played in the World Series, and the winner is the team who wins more games. In a way, this is what happens in reality; it’s as though the first team to lose 4 games (and thus cannot win more than the other team) forfeits the rest of their games. So, we can treat the actual World Series as a binomial experiment with seven trials. If WW is the number of games won by the Reds, the probability that the Reds win the World Series is P(W4)P(W4). Using the techniques from the last example, we get P(Reds win the series)=0.8002P(Reds win the series)=0.8002.

People in Mathematics

Abraham de Moivre

Abraham de Moivre was born in 1667 in France to a Protestant family. Though he was educated in Catholic schools, he remained true to his faith; in 1687, he fled with his brother to London to escape persecution under the reign of King Louis XIV. Once he arrived in England, he supported himself as a freelance math tutor while he conducted his own research. Among his interests was probability; in 1711, he published the first edition of The Doctrine of Chances: A Method of Calculating the Probabilities of Events in Play. This book was the second textbook on probability (after Cardano’s Liber de ludo aleae). De Moivre discovered an important connection between the binomial distribution and thenormal distribution (an important concept in statistics; we’ll explore that distribution and its connection to the binomial distribution in Chapter 8). De Moivre also discovered some properties of a new probability distribution that later became known as the Poisson distribution.

Check Your Understanding

You are rolling a 6-sided die with 3 orange faces, 2 green faces, and 1 blue face.
56.
If you roll the die 5 times and note the color showing on each roll, is this a binomial experiment?
57.
If you roll the die 5 times and count the number times you roll a green face, is this a binomial experiment?
58.
If you count how many times you roll the die until you get a blue face, is this a binomial experiment?
Suppose you’re rolling the same colored 6-sided die 10 times. Let O , G , and B represent the number of times the die lands with an orange, green, and blue side up, respectively. Find these probabilities (round to 4 decimal places):
59.
P ( O 7 )
60.
P ( G = 5 )
61.
P ( 1 B 4 )
In the following exercises, decide whether the described experiments are binomial experiments. For those that are not, explain why they aren’t.
1 .
A golfer practices putts from 1 foot, 2 feet, 3 feet, 4 feet, and 5 feet; “success” is defined as making the putt.
2 .
A game designer rolls a pair of dice 100 times and counts the number of times the sum is at least 10.
3 .
A student who is completely unprepared for a multiple-choice pop quiz guesses on all 10 questions. There are 4 choices for each of the first 5 questions and 5 choices for each of the last 5 questions. “Success” is defined as answering the question correctly.
4 .
A baseball player is practicing pitching; he throws pitches until he gets 50 strikes.
5 .
A statistician stops 20 college students at random outside a dining hall and notes their class year.
6 .
An employee at a bowling alley watches each patron’s first ball and counts how many are strikes over the course of his shift.
In the following exercises, you have an 8-sided die with 4 faces colored orange, 3 colored red, and 1 colored yellow. You’re going to roll the die 100 times: Let Y be the number of times a yellow face is showing, R be the number of times a red face is showing, and O be the number of times an orange face is showing. Find the given probabilities, rounded to 4 decimal places:
7 .
P ( O = 52 )
8 .
P ( R < 40 )
9 .
P ( Y 10 )
10 .
P ( O > 55 )
11 .
P ( R = 35 )
12 .
P ( Y < 5 )
13 .
P ( 45 < O < 55 )
14 .
P ( R 30 )
15 .
P ( Y 20 )
16 .
P ( O < 40 )
The following exercises are about series of games, where Team A faces Team B in a best-of series (like the World Series). Find the probability that Team A wins the series in each of the following scenarios:
17 .
P ( Team A wins a single game against Team B ) = 51 % best-of-5 series
18 .
P ( Team A wins a single game against Team B ) = 51 % best-of-7 series
19 .
P ( Team A wins a single game against Team B ) = 51 % best-of-15 series
20 .
P ( Team A wins a single game against Team B ) = 51 % , best-of-31 series
21 .
P ( Team A wins a single game against Team B ) = 51 % , best-of-101 series
22 .
P ( Team A wins a single game against Team B ) = 60 % , best-of-5 series
23 .
P ( Team A wins a single game against Team B ) = 60 % , best-of-7 series
24 .
P ( Team A wins a single game against Team B ) = 60 % , best-of-15 series
25 .
P ( Team A wins a single game against Team B ) = 60 % , best-of-31 series
26 .
P ( Team A wins a single game against Team B ) = 60 % , best-of-101 series
27 .
Give the table of the PDF for flipping a fair coin 5 times and counting the heads. Do the calculations without using technology.
28 .
Give the table of the CDF for flipping a fair coin 5 times and counting the heads. Do the calculations without using technology.
The following exercises are about the casino game roulette. In this game, the dealer spins a marble around a wheel that contains 38 pockets that the marble can fall into. Each pocket has a number (each whole number from 0 to 36, along with a “double zero”) and a color (0 and 00 are both green; the other 36 numbers are evenly divided between black and red). Players make bets on which number (or groups of numbers) they think the marble will land on. The figure shows the layout of the numbers and colors, as well as some of the bets that can be made.
A Roulette table. The table has 12 rows and 3 columns. The first four rows represent the first dozen. The fifth to eighth rows represent the second dozen. The last four rows represent the third dozen. The columns represent 2 to 1, 2 to 1, and 2 to 1. Two green pockets, 0 and 00 are at the top-right. Six green pockets on the left represent 1 to 18, even, red, black, odd, and 19 to 36. The data from the table row-wise are as follows: 1 (red), 2 (black), 3 (red); 4 (black), 5 (red), 6 (black); 7 (red), 8 (black), 9 (red); 10 (black), 11 (black), 12 (red); 13 (black), 14 (red), 15 (black); 16 (red), 17 (black), 18 (red); 19 (red), 20 (black), 21 (red); 22 (black), 23 (red), 24 (black); 25 (red), 26 (black), 27 (red); 28 (black), 29 (black), 30 (red); 31 (black), 32 (red), 33 (black); 34 (red), 35 (black), and (36 (red).
Roulette Table (credit: "American Roulette Table Layout" by Film8ker/Wikimedia Commons, Public Domain)
29 .
If a player bets $1 on red and wins, the player gets $2 back (the original $1 bet plus $1 winnings). What is the probability that the player wins more than they lose if the player bets on red on 5 consecutive spins?
30 .
What is the probability that the player wins more than they lose if the player bets on red on 15 consecutive spins?
31 .
What is the probability that the player wins more than they lose if the player bets on red on 30 consecutive spins?
32 .
What is the probability that the player wins more than they lose if the player bets on red on 100 consecutive spins?
33 .
What is the probability that the player wins more than they lose if the player bets on red on 200 consecutive spins?
34 .
What is the probability that the player wins more than they lose if the player bets on red on 1,000 consecutive spins?
35 .
What is the probability that the player wins more than they lose if the player bets on red on 5,000 consecutive spins?
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