Contents

When you take the FE Exam you are going to deal with Binomial Distribution problems on some level.

Most know this and, as a result, do what one naturally should do –

Get into solving practice problems.

However, when it comes down to it on the exam, many are fumbling their way through solving these types of problems – if they even attempt to solve them at all – why?

Because they are failing to gain a basic understanding of how to identify such a problem.

The goal right now shouldn’t be how to solve these problems with the Binomial Distribution formula.

That will come with time.

Right now, the main focus should be knowing how to identify a Binomial Probability Distribution problem – fast.

At the most fundamental level, you need to know what makes a Binomial Distribution problem different from other probability distributions.

Once you are solid on this front, you can then worry about the Binomial probability formula, binomial random variable and everything else that goes into working this discrete distribution.

So with all that –

## What is Binomial Distribution and its properties?

The binomial distribution is a probability density function that we use as engineers to help us establish the statistical likelihood that either one of two mutually exclusive events will occur, given a set of parameters and assumptions.

The main property of a Binomial Distribution is, welp:

The “Binary” nature of it – the fact that it measures around two possible outcomes, true or false, success or failure, yes or no.

Other properties of a Binomial experiment include:

- There are ‘n’ number of independent trials or a fixed number of n times repeated trials.
- The probability of success or failure remains the same for each trial.

The most rudimentary example of a Binomial experiment would be that of a simple fair coin toss.

Binomial distributions can be summed up as representing **the probability for x successes in n trials, given a success probability p, for that event occurring in each trial.**

## How do you know if it’s Binomial Distribution?

When taking the FE Exam, we need to know quick if we are reading a Binomial Distribution, Poisson Distribution or a Normal Distribution problem- amongst others.

So how can we know if it’s a Binomial Probability problem quickly?

As you are reading through the problem statement be asking yourself these questions:

- Am I reading a scenario that involves a fixed number of independent trials?
- Does this experiment produce only binary results? Success or failure, true or false, yes or no?
- Is the probability of success in each trial remain constant?

These are the core elements of a Binomial experiment, simple enough.

However, we can expect that the NCEES will attempt to hide them in the midst of other information – whether relevant or irrelevant.

For example, you may be asked to find the probability of a certain number of successes in a set number of trials, or to calculate other statistics related to the binomial distribution.

However, if the problem meets the core criteria, it is likely a binomial distribution problem.

## What is difference between Binomial and Poisson distribution?

Knowing the characteristics of a Binomial Distribution problem will get you much futher identifying than most (and quicker), but knowing the difference between distributions altogether will ensure reliability in your call.

So what is the difference between Binomial and Poisson distribution?

Binomial distribution describes the distribution of binary data from a finite sample.

It gives us the probability of getting x events out of n trials.

Poisson distribution on the other hand describes the distribution of binary data from an infinite sample.

It gives us the probability of getting r events in a population as a whole.

As you are reading through a problem statement, they might read much the same, with one glaring difference.

A Binomial experiment has a *fixed number of trials* while a Poisson experiment has an *infinite number of trials.*

## What is difference between Binomial and Normal distribution?

What makes Binomial distribution different from Normal Distribution?

The main difference between a binomial and normal distribution is the type of data that is being modeled.

A binomial distribution models data that consists of the number of successes in a fixed number of trials, where each trial results in either a success or failure, and the probability of success is constant.

A normal distribution, on the other hand, models data that is continuous and normally distributed – the data tends to cluster around the mean and has a bell-shaped distribution.

So again –

Like Poisson Distribution, Normal Distribution deals with an infinite (continuous) sampling.

When reading through a problem statement, you know to look for the identifying characteristics of a Binomial experiment:

- Fixed number of trials
- Binary results (success/failure, etc)
- A probability of success that is constant for each trial

For standard normal distribution problems, you will read scenarios involving *continuous data* and *requests to find probabilities associated with normal approximation,*.

Additionally, the problem statement may include other statistical data like the population mean, variance and standard deviation which are used in the probability mass function.

## What is binomial distribution with example?

A binomial distribution is a type of probability distribution that shows the likelihood of a given number of a certain event occuring in a fixed number of trials.

As mentione, the most basic example of a binomial experiment is that of flipping a fair coin 10 times and counting the number of times it lands on heads.

In this experiment, we have a **fixed number of independant trials (10)**, each with a **binary result (heads or tails)** and a **constant probability of success, .5.**

An example in the realm of engineering would be one where an engineer is testing the strength of a new type of steel alloy.

The problem statment may tell us that the probability that a sample of the alloy will pass a strength test is 0.8, and with that, ask us to determine the probability that out of 10 samples, exactly 7 will pass?

What tends to confuse in problem statements like this is the way they are asking the question – *how many will pass in a certain number of trials.*

However, if we stay focused on the core characteristics of a Binomial experiment we would be able to conclude quickly what we are reading.

In this problem, core elements tell us that we have a **fixed number of trials (10)**, each with a **binary result (pass or fail)** and a **constant probability of passing, .8.**

Knowing this information, we could move forward with deploying the Binomial formula to determine the correct answer.

## What is an example of a Binomial Distribution problem?

An example Binomial Distribution problem in the world of engineering may read something like this:

*A sample containing 400 items is taken from the output of a production line. Defective items occur randomly and independently at a rate of 1.6% of the population. The probability that the sample will have no more than 2.5% defective items is most close to what?*

The first step to solving this problem is knowing quickly that it is indeed a Binomial Distribution problem.

Let’s read over the problem statement again.

- Does this scenario involve a fixed number of trials? Yes, 400 items
- Does the experiment produce only binary results? Yes, either a defective or nondefective item.
- Is the probability of success in each trial constant? Yes, 1.6% defect rate

The core characteristics of a Binomial experiment are present and we can move forward with determining the cumulative probability.

You shouldn’t expect to see a problem that would require such a long solution on the FE Exam, but the steps would be as follows:

- Identify the core data being given in the problem statement. We are told that there are
**400 items (n)**, each with an**expected failure rate of 1.6% (probability p = .016)**and we want to know**the probability of having no more than 2.5% defective items in our finite sample.** - With this data given, we can determine that our rate of a non-defective item would be 98.4% (q = 1 – p = .984) and that we want no more than 10 defective items (X = 400 x .025 = 10)
- From here, use the binomial probability formula provided to us by the NCEES for Binomial Distribution on page 66 of the NCEES FE Reference Handbook and solve for the probability that there would be no more than 10 defective items in this particular sample, P(X ≤ 10).
This would be a lenghty process, but you would do this one by one, using the binomial distribution formula to determine the probability for 0 defects, P(0), the probability for 1 defect, P(1), the probability for 2 defects, P(2) and so on up to 10 defects, P(10). Once you have the individual values, you would then add all those probabilities together to get the cumulative probability.

This illustrates the reason why we train our Prepineer students how to hack these types of problems on the FE Exam – you just can’t afford to spend the time it will take to solve, nor do you need to.

At Prepineer, we show you a much more efficient (and reliable) way to define the cumulative probability by deploying the binomial distribution formula in a fraction of the time it is taking others:

Hacking Binomial Distribution problems with your NCEES approved calculator

## Practice identifying Binomial Distribution problems

With all of our groundwork now established, let’s run through a set of problems to get in some practice.

Whether we get Binomial Distribution, Normal Distribution, Poisson Distribution, or even Chi-squared Distribution problems on our FE Exam, we need to be able to identify them fast and move to successfully solving them.

Let’s get into it.

###### 1: An engineer is designing a new type of light bulb that is supposed to last for 1000 hours. It is known that the probability of a light bulb lasting for its full 1000 hours is 0.95. The probability that out of 10 light bulbs, exactly 8 will last for their full 1000 hours is most close to.

The **first step** in identifying whether or not this is a Binomial Distribution problem is to **determine if it includes a fixed number of trials.**

The answer in this case is yes, they will be pulling *a finite sample of 10 light bulbs to test.*

Now the **second step** is to ask ourselves, in testing these 10 light bulbs, are their **only binary results that can be obtained?**

The answer to this is yes, the light bulb *will either last the full 1000 hours or it won’t. *

We are well on our way to identifying this problem, but before we do, ask this last question – **does each trial have a constant probability of success?**

It does, the problem states that their is *a 95% chance of the light bulb lasting the full 1000 hours.*

Therefore, we can conclude that this is indeed a Binomial Distribution problem.

###### 2: An engineer is testing the strength of a new type of steel alloy. She knows that the strength of the alloy follows a normal distribution with a mean of 80,000 psi and a standard deviation of 2,500 psi. She has only a small sample of 10 steel alloy samples to test, calculate the probability that the mean strength of the sample will be at least 75,000 psi.

The **first step** in identifying whether or not this is a Binomial Distribution problem is to **determine if it includes a fixed number of trials.**

The answer in this case is no.

A core characteristics of a binomial distribution problem is that *it describes the outcome of a number of independent “success or failure” events*, whereas this problem is describing *a continuous variable* (the strength of the steel alloy) that follows a normal distribution.

Therefore, we can conclude that this is not a Binomial Distribution problem.

###### 3: An engineer is designing a new type of solar panel. They know that the panel’s power output follows a normal distribution with a mean of 300 watts and a standard deviation of 20 watts. What is the probability that a randomly selected panel will have a power output of at least 320 watts.

Are we being given a scenario that involves a fixed number of trials?

We aren’t, the power output of the solar panel is a continuous variable that is stated to follow a normal distribution.

Therefore, we can conclude that this is not a Binomial Distribution problem.

###### 4: An engineer is testing the reliability of a new type of electronic component. They know that the probability of a component failing a stress test is 0.1. What is the probability that out of 10 components, exactly 3 will fail the stress test?./h6>

Does this scenario involve a fixed number of trials? Yes, testing 10 components.

Does the experiment produce only binary results? Yes, pass or fail.

Is the probability of success in each trial constant? Yes, the probability of a component failing the stress test is 0.1.

Therfore, the core characteristics of a Binomial probability distribution are present and we can move on to solving the problem.

###### 5: An engineer is testing the efficiency of a new type of solar panel. They know that the probability of a panel generating at least 90% of its rated power output is 0.8. What is the probability that out of 10 panels, exactly 9 will generate at least 90% of their rated power output?

Does this scenario involve a fixed number of trials? Yes, testing 10 panels.

Does the experiment produce only binary results? Yes, it either produces 90% of it’s rated power or it doesn’t.

Is the probability of success in each trial constant? Yes, the probability of a panel generating at least 90% of its rated power output is 0.8.

Therfore, the core characteristics of a Binomial probability distribution are present and we can move on to solving the problem.

###### 6: An engineer is tasked to optimize the design a new type of battery. They know that the battery’s capacity is normally distributed with a mean of 2,000 mAh and a standard deviation of 100 mAh. If a sample of 6 batteries are tested, the probability that the sample is significantly different from the expected mean of 2,000 mAh is?

This problem is not a binomial experiment because it involves a continuous variable (battery capacity) and is looking at the probability of the sample being significantly different from the expected mean, rather than the probability of a specific number of successes or failures occuring in a series of independent trials.

###### 7: A manufacturer is producing a batch of 100 widgets. The widgets have a 0.01% failure rate, meaning that on average, 1 out of every 10,000 widgets produced will be defective. The manufacturer wants to know the probability of producing exactly 5 defective widgets in this batch.

The first step in identifying whether or not this is a Binomial Distribution problem is to determine if it includes a fixed number of trials.

The answer in this case is yes, they are producing a batch of 100 widgets.

Now the second step is to ask ourselves, in producing these widgets, is their only binary results that can be obtained?

The answer to this is yes, the widget will either be defective or not defective.

We are well on our way to identifying this problem, but before we do, ask this last question – does each trial have a constant probability of success?

It does, the problem states that their is a .01% chance that the widget will be defective.

Therefore, we can conclude that this is indeed a Binomial Distributions problem.

###### 8: An engineer is designing a new type of airplane wing. The wing is made up of many small components, and the engineer knows that each component has a 0.1% chance of failing. If the wing has 1,000 components, what is the probability that at least 150 of them will fail?

Does this scenario involve a fixed number of trials? Yes, the wing has 1,000 components.

Does the experiment produce only binary results? Yes, each component either fails or it doesn’t.

Is the probability of success in each trial constant? Yes, the probability of each component failing is 0.1%.

Therfore, the core characteristics of a Binomial experiment are present and we can move on to solving the problem.

###### 9: An engineer is testing a new type of rubber for a truck tire. The tire has a 95% probability of not failing under normal driving conditions, and a 5% probability of failing. The engineer wants to know the probability of the tire failing at least once in a test drive that simulates 10 years of driving (assume the tire is driven on for 8 hours a day, every day).

Does this scenario involve a fixed number of trials? Yes, a trial is a day for 10 years, driving under normal driving conditions.

Does the experiment produce only binary results? Yes, the tire either fails or it doesn’t.

Is the probability of success in each trial constant? Yes, the probability of the tire not failing is 95%.

Therfore, the core characteristics of a Binomial experiment are present and we can move on to using the this probability mass function to solve the problem.

###### 10: A new type of robot arm is nearing completion in design. It is known that this new arm has a 99% probability of functioning correctly, and a 0.01 probability of malfunctioning. If the normal lifespan is 10,000 hours, and that it is used for 8 hours a day, every day. What is the probability of the robot arm malfunctioning at least once during its lifespan.

Does this scenario involve a fixed number of trials? Yes, 10,000 hours of lifespan.

Does the experiment produce only binary results? Yes, the robot arm either malfunctions or it doesn’t.

Is the probability of success in each trial constant? Yes, the probability of the robot arm malfunctioning is 1%.

Therfore, the core characteristics of a Binomial experiment are present and we can move on to solving this discrete distribution.

## Why you need to identify Binomial Distribution problem fast

There are many reasons why students are struggling to pass the FE Exam.

The material is only part of the story.

More so the experience for many is that they are just flat out running out of time on the actual exam.

If we aren’t able to identify the types of problems we are being presented quickly and efficiently, we are significantly reducing our chances of success on the FE Exam.

This is something we can’t allow.

Without a doubt, you will see Binomial Distributions on your exam.

Question is, **will you know what you are looking at?**

We believe you will, as long as you remember to filter each Engineering Economics problem statement using these questions:

- Is the scenario one that involves a fixed number of repeated trials?
- Does the experiment produce only binary results? Success or failure, true or false, yes or no?
- Is the probability of success in each trial remain constant?

These are the core elements of a Binomial experiment.

If the answer to these question are *yes*, then you are looking at a Binomial Distribution problem – it’s now time to deploy the binomial formula and solve it.

Or better yet, hack it using the method found here.

## What’s the next step?

If you feel you have **wasted time** trying to study *but haven’t moved the needle* any closer to passing the FE Exam.

If you feel you are **just guessing** on what needs to be done *instead of having a clear picture* of what it will take.

If you feel **passing the FE is far past your capabilities.**

Then know this.

It’s not and –

Passing the FE Exam doesn’t have to remain a mystery.

Sadly though, many are doubting that they can get it done, and in turn, *are giving up on themselves and their careers.*

I don’t want that to be you – if I offered to show you a system that has worked for thousands in your same shoes, *would you take it?*