Learn more about binomial probability and its practical applications, from determining airline overbooking strategies to calculating probabilities through Excel functions. This article demystifies binomial distributions and illustrates their use in real-world scenarios.
Key Insights
- Binomial probability calculates the likelihood of achieving a specific number of successes within a finite number of independent trials, each with a consistent probability of success or failure.
- The article demonstrates practical applications such as predicting passenger attendance for airline flights, revealing a mere 0.00351 percent chance that all 200 booked passengers would show up, influencing airline overbooking strategies.
- Excel’s binomial functions, including BINOM.DIST, BINOM.DIST.RANGE, and BINOM.INV, simplify complex calculations and help answer questions like determining the ideal ticket count (196 seats) for a 99 percent probability against flight overbooking.
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Binomial probability. Binomial probability allows you to calculate the probability of a certain number of successes occurring in a given number of trials. So there are three fundamental aspects of binomial probability that you need to take into account in order for this calculation to work.
The first is n independent trials must occur. So that's the first aspect. Trials.
The other aspect is each trial is either a success or failure. It could be either one. Success is one.
Failure is zero. Each trial has a constant probability of success or failure depending on what you chose in the prior criteria. So briefly to summarize, binomial distribution describes the distribution of binary data, success or failure, from a finite sample.
Thus it gives the probability of getting a number of desired events out of trials run. Binomial means two or bi means two. Nominal number, success or failure, yes or no.
Finite sample means a trial experiment is conducted a certain number of times. When to use binomial distribution? When given up an exact probability and you need to find the probability of an event happening a certain number of times out of X trials. For example, three times out of ten.
We have the mathematical approach here. But as usual, Excel has a function that is going to do most of the heavy lifting for you so you don't have to understand the math that's involved. But if you actually wanted to take a look at the math, we have an example here listed in practice one.
So you can look at this on your own time. Let's now move over to the Excel version of calculating what we did mathematically here. The function we're going to use is called binomial distribution.
It has four arguments, the number of trials, well the number of successes first, the number of trials, the probability, and whether or not this is related to cumulative or referring to a single point. A basket contains three toys, one red, one blue, one green. If three toys are randomly selected with replacement, what is the probability that none will be red? How many trials? One trial.
Well, that makes sense. Red, the possibility of picking up red is 33 percent. So the probability of not picking red is 66.7 percent the first time.
Now, what if I try this a second time? Well, my probability goes down. Now it's only 44 percent. It's under 50 percent.
The probability that I will not pick up red goes down to 44 percent. How about three times? Well, now it's down to 29 percent. So there's now practically a 70 percent chance that I will pick up the red toy.
Why? Because it depends on all my prior attempts and my ability to not select it. I'm, in a way, pressing my luck. So I'm going to change this back to one.
Let's take a look at a practical example related to the airline industry. So assume there's a 95 percent probability that each passenger that books a ticket actually shows up for their flight and that a flight has exactly 200 seats. What is the probability that all 200 passengers will show up? So I want to calculate this using binomial distribution.
So I will type equal binomial distribution. Now, the number of successes is 200. I would like 200 passengers to actually show up.
That's what I'm anticipating. And that's what there is a 95 percent chance of happening based on the probability. Now, the number of trials is 200.
So each of those 200 passengers represents 200 trials. I enter a comma. The probability is 95 percent.
I need to decide whether or not this is going to be a probability mass function, which refers to a single point in my statistics, or whether or not this is cumulative. And I'm looking for values starting from a certain point and then going down to zero and below. In this case, it's going to be false.
I'll press ENTER. There is a 0.00351 percent chance that all 200 passengers will show up. So this now presents an opportunity for the airline industry, knowing that this is the case, to perhaps sell more tickets than there are seats.
Because if the probability is 0.003 that all 200 passengers will show up, that may mean the airline is going to have empty seats. So rather than have empty seats, they will sell more tickets than there are seats in hopes that all the seats are filled. Now, if more people show up, well, they can give them credit for another flight in the future.
Maybe the cost of paying for a hotel and issuing that credit is going to be less than not having the seats filled. So let's say we wanted to find out what is our best percentage of people actually showing up for the flight. This is going to be the mean on the bell curve.
And we actually have some calculations on the right that display this. We're going to say, well, maybe let's be a little conservative. Instead of expecting 200 people to show, how about 190? So I'm going to type equal binomial distribution.
The number of successes is going to be 190. The number of trials is going to be 200. The probability is still 95%.
Again, I'm looking for a point, a specific point on the bell curve graph. So I'm going to choose false for probability mass function. I'll press ENTER.
12%. In this case, 12% is going to be the best that we're ever going to get in terms of people arriving for the flight. We have it listed here.
This is our bell curve. Anything else will be less on either side. So that answers two questions.
We have a different type of question. Maybe we're not encouraged by the low percentage. So for exercise three, we're going to ask what is the probability that a range of 1 to 190 passengers show up? Now here, we're looking for multiple values.
We're not looking for a single value. We're not going to choose the probability mass function, which is false. We'll actually choose true, which will cumulatively look at values from 1 to 190.
Now I'll type equal binomial distribution. Number of successes is 190. I'll enter a comma.
Number of trials is 200. The probability is still 95%. But this time, I'm going to say true and choose the cumulative distribution function.
That means we'll start at 190 and go down to zero. And so we're looking for a percentage for the number of people showing up as a range between 1 and 190. I'll press ENTER.
54%. That makes sense. That is a strong percentage.
If we sell 200 tickets, we expect anywhere from 1 to 190 people to show up. All right. So again, this is how the airline industry can hedge their bets in terms of being able to fill all the seats and sell tickets based on the probability of people showing up.
Let's take a look at some additional calculations. We do have two additional binomial distribution functions, range and inverse. So let's take a look at range.
Here is the question we're looking to answer. If the airline oversells the plane by five seats, what is the probability that the flight will be overbooked? We decide based on our calculations that we're going to sell more tickets than we have seats. But now the percentage we want to know about is the percentage that we might overbook.
All right. So I'm going to type equal binomial distribution, but make sure to choose range. It's easy to choose the very first binomial distribution function, the one we've been using all this time.
Now the number of trials is going to be 205. Again, we have a 95 percent probability. And now is the opportunity to list the range.
The range is going to be from 201 to 205. That's what we're looking to overbook by. So I'll select the first value comma, and then I'll select 205 close parentheses when I press ENTER.
Two point two four percent. So it's a pretty safe bet that I can get away with booking 205 seats. Or selling 205 seats if I only have 200 seats available.
There's not that much chance that I will overbook. Now, we have another calculation. How many seats should be sold to ensure a 99 percent probability that the flight will not be overbooked? So the inverse of this is we definitely don't want to overbook the flight.
So what's the minimum amount of tickets we can offer up for sale so that we don't overbook the flight? So that's going to be equal to binomial dot inverse. You're going to select the number of trials, which is 200. The probability, which is 95.
The alpha is going to be 0.99. We want to ensure a 99 percent probability that the flight will not be overbooked. I'll press ENTER, and that is 196. If we sell 196 tickets, we definitely will not overbook the flight.
That's based on a probability of 95 percent. So that's why we're getting 196. All right.
So for binomial probability, we took a look at how we could calculate the number of successes based on a certain number of trials on an agreed upon probability of either the success or failure. We do have some charts here on the right that visualize the information that we were working with. When you're using false, you're referring to a point, just like we talked about earlier with the bell curve.
But if you're using true, which stands for cumulative, then you're selecting that value and all the values underneath. So that is it for binomial probability. Thank you for watching.