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There are a variety of experiments whose outcomes can be reduced to success or failure. This lesson will explore how real-life situations that satisfy specific conditions can be modeled as binomial experiments. Additionally, methods of determining the probability of a certain number of successes in binomial trials will be presented.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.
Understanding Probability

Understanding Descriptive Measures

Understanding Types of Data

Other Recommended Readings

Explore

Investigating Distributions Using Experiments

The Galton board is a device patented by Sir Francis Galton. It consists of a set of balls that are dropped from the top of the board. As the balls fall, they move to the left or right every time they bounce off of the pegs embedded in the board until they land in one of the bins at the bottom. Each path option has the same probability.
A Galton Board with a set of marbles ready to be dropped.
Based on the outcomes, try to ask the following questions.
  • Which bins would be expected to collect the most balls?
  • About how many balls are expected to land in each bin?
  • How many paths branch out from each peg?
Discussion

Analyzing the Probabilities of the Values of a Random Variable

A random variable assigns a numerical value to an outcome of a probability experiment. In many situations, it is important to know how likely it is that a random variable will take a specific value. This can be represented by listing or graphing the probability of each value of a random variable. This is called a probability distribution.

Concept

Probability Distribution

A probability distribution of a random variable is a function that gives the probability of each outcome in the sample space. It can be represented by tables, equations, or graphs. A probability distribution needs to satisfy two conditions to be valid.

  1. The probability of each value of is greater than or equal to and less than or equal to
  2. For a discrete variable, the sum of the probabilities of all possible values of is
A probability distribution can be either discrete like the geometric distribution and the binomial distribution, or continuous like the normal distribution.
Discrete and Continuous Distributions
If a probability distribution is based on mathematical models and assumptions, then it is called a theoretical probability distribution. On the other hand, an experimental probability distribution is determined by conducting an experiment.

Example

Consider the roll of a pair of standard dice. Let be the random variable that represents the sum of the two dice. By the fundamental counting principle, since rolling each die has possible outcomes, there are a total of possible results. Additionally, the possible values of are integers from to

A table that represents the theoretical probability distribution of will now be created. Frequencies represent the number of dice roll results that add up to the given values of the random variable The frequency is divided by to determine the theoretical probability of each outcome.

Frequency
The probability distribution of can also be represented by a bar graph. The possible outcomes of are marked on the horizontal axis and their probabilities are presented on the vertical axis.
Sum of Two Dice
In this example, each possible value of the random variable can be associated with its corresponding probability because it is a discrete random variable. This is also why the bars of the probability distribution must be separated.
Example

Determine the Theoretical and Experimental Probability Distributions of a Coin Toss

Izabella is a big soccer fan. One weekend, she invited her friend Dylan to watch a world championship match together at her house. During the coin toss ceremony, Izabella asked Dylan about the number of heads they will obtain if they toss a fair coin four times.

Tv-soccer.jpg

Let be a random variable that represents the number of heads in four coin flips. Help Izabella and Dylan solve the following problems and determine whether they can predict the number of times the experiment results in heads.

a Construct a table to describe the theoretical probability distribution of
b Graph the theoretical probability distribution of
c Izabella and Dylan conducted an experiment of tossing the fair coin four times. They repeated this experiment times and recorded the data in a tally sheet.
Number of Heads, Tally Frequency

Use this data to find the experimental probability of each possible value of

d Graph the experimental probability distribution of

Answer

a
b
The theoretical probability of obtaining a specific number of heads in four coin flips. The vertical axis represents probability, ranging from 0 to 0.5 with increments of 0.1. The horizontal axis displays the number of heads, labeled from 0 to 4, corresponding to the possible outcomes. Each bar represents the probability of getting that exact number of heads in four flips.
c
d
Experimental Probability Distribution of the Four-Coin Toss.

Hint

a Think about the possible outcomes of tossing a coin four times. How can the outcomes be represented?
b In a discrete distribution, the bars on the graph will be separated.
c Divide the frequency of each outcome by the total number of trials.
d The horizontal axis will represent the possible outcomes and the vertical axis will represent the probability of each outcome.

Solution

a The theoretical distribution of will be determined by using a table. Observe the following simulation and think about the possible values for Keep in mind that there are exactly four coin flips.
Simulation of the toss of a fair coin.
Because represents the number of heads in four coin flips, the possible values of the variable are and To analyze the outcomes where each value is obtained, mark heads as and tails as Then, write all possible outcomes for each number of heads and count them.
Possible Outcomes
Frequency
Before calculating the theoretical probability of each value, find the total number of possible outcomes.
Now divide the frequency of each outcome by the total number of possibilities to calculate the probability that the random variable takes the specific value
Frequency

This table describes the theoretical probabilities associated with tossing a fair coin four times. Since only the theoretical probability is required, the Frequency column can be skipped.

b To graph the distribution, each possible outcome will be represented on the horizontal axis and the probability of each outcome on the vertical axis. Since only and instances of heads are possible — it cannot appear times out of tosses, for example — the probability distribution is discrete and the bars on the graph will be separated.
Distribution Probability of the Four-Coint Toss.
c Consider the tally sheet prepared by Izabella and Dylan when conducting trials of the experiment consisting of tossing a fair coin four times.
Number of Heads, Tally Frequency

Calculate the experimental probability of each possible outcome by dividing its frequency by the total number of trials,

Number of Heads, Tally Frequency

Since only the experimental probability is required, the Tally and Frequency columns can be skipped. Next, write the table horizontally.

d Follow a similar method as in Part B to graph the probability distribution. Again, the horizontal axis will represent the possible values of while the vertical axis will represent the probability of each outcome.
Distribution Probability of the Four-Coint Toss.
From the graph, it can be seen that the outcome with the highest probability of occurring is This implies that will be the most common outcome as the experiment is continued to be performed. This matches what the theoretical probability calculated previously.
Discussion

Can the Experimental Distribution Estimate the Theoretical Distribution?

The expected value of a random variable is the average of the possible outcomes of a random variable. It is used to describe the center of a probability distribution. For a discrete random variable, the expected value is given by the weighted mean.

In this formula, represents a specific outcome, corresponds to the associated probability of and is the number of all possible outcomes. According to the law of large numbers, when considering a sequence of random variables, its average tends to the expected value under specific conditions.

Rule

Law of Large Numbers

Let be independent random variables with an identical probability distribution and be the mean of the random variables.
The independence of random variables means that the events that different random variables take particular values are independent. If represents the expected value of the law of large numbers states that tends towards

This notation means that for a sufficiently large number of observations — or by repeating the experiment a large number of times — the mean will approximate the expected value This law can be used to find the theoretical probabilities of the outcomes of an experiment by using the experimental probabilities. Consider the flipping of a fair coin experiment.
flipping of a fair coin
Suppose it is unknown that the probability of landing on either heads or tails on a single trial is For a few coin tosses, the experimental probability might not come anywhere near
An applet showing how the experimental probabilities of the possible outcomes vary after each trial.
However, as the number of trials increases, the proportion of heads and tails outcomes will be closer to their theoretical probabilities. Suppose that this experiment is repeated or even times, then consider the event that the outcome of the coin toss is tails.
An applet showing how the experimental probability of tails gets closer to its theoretical probability as the number of trials increases.
This law helps understand why casinos always win. Someone can be lucky and win a certain amount of times, but in the long run, the casino's earnings will converge to a predictable percentage, the expected value. The house always wins.
Discussion

How is the Variation of a Probability Distribution Measured?

The expected value is commonly used with a measure of variation such as the variance or standard deviation to determine how outcome will differ from the expected value.

Concept

Variance of a Random Variable

The variance of a random variable describes how far from the expected value the outcomes of a random variable are likely to be. To calculate the variance, begin by determining the deviation of each possible outcome — the difference between and
The variance is the total sum of the products of the deviation of each outcome and its corresponding probability
The variance is denoted by because it is the square of the standard deviation
Concept

Standard Deviation of a Random Variable

The standard deviation of a random variable is a measure of variation that describes how spread out the outcomes of a random variable are from its expected value The standard deviation is represented by the Greek letter — read as sigma — and is given by the square root of the variance of

In this formula, is a specific outcome and is the probability of

Example

Let be the random variable representing the number of cars sold on a given day in a car dealership. The table below shows the probability distribution of

Use the probability distribution to calculate the standard deviation of Begin by finding the expected value.
Evaluate right-hand side

Next, the variance of will be calculated using a table of values. To do so, calculate the square of each deviation — the difference between each possible value of and Then multiply the square deviation of each value by its corresponding probability.
Variance

Finally, calculate the square root of the variance to get the standard deviation of

Variance vs Standard Deviation

The standard deviation is preferred over the variance because taking the square root of the variance gives the standard deviation the same units as This makes it possible to compare the possible outcomes relative to the expected value.
Example

Using Standard Deviation to Make a Decision

Izabella's aunt Magdalena owns a clothing store. She needs to increase stock in her shop and plans to invest in one of the two collections that were offered to her by well-known brands. Each brand claims that they have a great expected rate of return. Their probability distributions are described below.

Dylan and Izabella want to help Magdalena make the best decision. They decided to use their recently acquired knowledge about the expected value and standard deviation of a probability distribution to analyze the offers. Help them answer the following questions and give the best advice to Magdalena.

a Pair each description with its corresponding measure.
b Which investment should Izabella and Dylan advise Magdalena to choose?

Hint

a The expected value is the total sum of the products of every possible value of the random variable and its corresponding probability.
b Compare the expected values and the standard deviations of the distributions.

Solution

a To match each description with its corresponding value, find the expected value and the standard deviation of each probability distribution one at a time.

Collection I

The expected value of a discrete random variable is given by the total sum of the products of every possible value and its associated probability Let represent the profit of the first collection. Note that each loss will be represented by a negative value.
The expected value can now be used to find the standard deviation of the probability distribution. Consider the formula for the standard deviation.
To apply this formula, calculate the square of each deviation — the difference between each value and the expected value — and then multiply it by its corresponding probability. This process will be shown in a table.
Sum of Values
The sum of the values of the last column represents the variance of the probability distribution. The standard deviation will be found by taking the square root of

Collection II

Follow a similar procedure as before to calculate the expected value for Collection II.
Now use a table to calculate the variance of the distribution.
Sum of Values
The square root of the variance will be calculated to find the standard deviation of the probability distribution of Collection II.

Conclusion

The expected value and the standard distribution of each probability distribution have been calculated. The following table summarizes these measures.

Measures of the Probability Distributions
Expected Value of Collection I
Expected Value of Collection II
Standard Deviation of Collection I
Standard Deviation of Collection II
b Consider the expected value of each distribution.
The expected values do not give much information on their own because they are very close. This means that the expected profit of each collection is similar. Next, compare the standard deviations to determine which distribution has more variability.
The standard deviation of Collection II is almost four times the that of Collection I, which implies that the expected value of Collection II will have about four times the variability of Collection I. Since Collection II is riskier, with a high chance for both gains and losses, Izabella and Dylan should advise Magdalena to invest in Collection I.
Discussion

Binomial Experiments

The outcomes of many experiments can be reduced to two possibilities, success or failure. If two more conditions are satisfied, these experiments can be modeled by a binomial experiment.

Concept

Binomial Experiment

A binomial experiment is a probability experiment that has the following three properties.

  1. There is a fixed number of independent trials.
  2. Each trial has exactly two possible outcomes — success and failure.
  3. For each trial, the probability of success is constant.
Details of each property can be seen in the applet by selecting the corresponding number.
properties of a binomial experiment
The number of successes in a binomial experiment is a random variable that has a binomial distribution.

Example

One of the most well-known examples of a binomial experiment is flipping a coin. There are only two possible outcomes, heads and tales. Either heads or tails could be considered a success, depending on the situation. The outcome of one toss does not impact the probability of the outcome of the next trial.
flipping a coin

Extra

An Experiment Reducible to a Binomial Experiment

Note that many probability experiments can be reduced so that they satisfy the conditions of a binomial experiment. In case of rolling a die, there are six possible outcomes. However, they can be divided into two groups — even numbers and odd numbers, for example.

Partitioning possible outcomes of rolling a die (1, 2, 3, 4, 5, 6) into two groups. Even numbers (2, 4, 6) are shown in the left bottom corner, while odd numbers (1, 3, 5) are shown in the right bottom corner.
After grouping the outcomes, there are two possible results of each trial — rolling an even number or an odd number. The probability of rolling a number from either group is constant throughout the trials, and the result of rolling the die is not affected by the previous results.
rolling a die
Therefore, rolling a die can also be considered as a binomial experiment as long as success and failure are clearly defined.
Example

Classifying Whether Experiments Are Binomial

To better understand binomial experiments, Dylan and Izabella listed some examples of experiments. They want to determine whether the experiments can be modeled as binomial experiments. Which of these situations are examples of binomial experiments?

Hint

Is there a fixed number of trials for each experiment? How many outcomes are possible? Does the probability of each outcome remain constant for each trial? Are trials independent?

Solution

Start by recalling the conditions that a binomial experiment should satisfy.

  1. There is a fixed number of independent trials.
  2. Each trial has exactly two possible outcomes — success and failure.
  3. For each trial, the probability of success is constant.

Analyze each situation one at a time to see if it follows all of the conditions of a binomial experiment.

The Scratch-off Cards

This situation has eight trials of selecting one scratch-off cards at random.

Scratch-off card
Each card could win a prize or not, which means there are two possible outcomes for each trial. Moreover, the probability of success, which is winning a prize, is or for every card.
Finally, the trials are independent because scratching one card does not affect the probability that any of the other cards reveals a prize. Therefore, this situation represents a binomial experiment.

Heights of People

Note that height can vary for every people surveyed.

People of different heights

Because there are possible answers, it is likely that more than different outcomes will occur. This means that this situation does not represent a binomial experiment.

Rolling a Die

In this case, rolling a can take only one or many more trials.
A die
It cannot be known how many rolls it will take until a comes up. Therefore the number of trials is not fixed. This means that this experiment does not represent a binomial experiment.

Blood Type O

This situation has a fixed number of trials because it involves asking people if their blood type is O.

People surveyed about their blood type
Each trial has only two possible solutions, blood type O or another type. Moreover, the probability of having blood type O in each trial is which represents the probability of success.
Finally, because the answer of any person surveyed does not affect the probability that other people have O type blood, each trial is independent. This means that this situation is a binomial experiment.

Conclusion

All four situations have already been analyzed and the results are summarized in the following table.

Experiment Is It a Binomial Experiment?
The Scratch-off Cards
Heights of People
Rolling a Die
Blood Type O?
Discussion

Probability of Successes in Binomial Trials

Because binomial experiments can simplify many complex situations, it is essential to determine how likely it is to obtain a specific number of successes out of trials in a given experiment. Also, the expected value, or center of the distribution, will be presented.

Concept

Binomial Distribution

The binomial distribution is the probability distribution that describes the number of successes out of binomial trials. The trials must satisfy three conditions.

  1. There is a fixed number of independent trials.
  2. Each trial has exactly two possible outcomes — success and failure.
  3. The probability of success is constant for each trial.

Let be a random variable representing the total number of successes among trials. The possible values of are The binomial probability formula can be used to determine the probability of successes among trials

In this formula, is the binomial coefficient and and are the probabilities of success and failure, respectively. Additionally, the expected value of can be determined by the product of the number of trials and the probability of success

This means that the expected number of successes in trials is given by

Example

Consider the experiment of drawing card from a standard deck of cards with replacement.

Cards from the As to K
If drawing a diamond is considered a success, let be the number of diamonds drawn. Since there are diamond cards in a standard deck, the probability of success in each trial will be The probability of failure will be Suppose the experiment is repeated times. Then, the binomial distribution with can be determined.
In this situation, the possible values of are and The distribution of the random variable can be obtained by evaluating this formula for each of these values.