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What is Empirical Probability?

Author: Adam
Published:September 14, 2022
16 mins 15 secs
What is Empirical Probability?

The world around us is full of interesting data and statistics. We can use empirical probabilities to make educated guesses about the world around us and make better choices in our lives. Empirical probability is a fun and valuable way for statisticians and mathematicians to think about the real world!

Probability is a numerical representation of the potential outcomes of an event. It can be expressed as a fraction or a decimal or written as a percentage. The possible results for any given situation should add up to 100%.

If they do not, it is incorrect and considered an experimental error. Probabilities are usually based on previously recorded encounters in similar situations. From these recorded values, probabilities are created through mathematical formulas that think how many total possibilities exist, how many possible outcomes there were, and how often each outcome occurred in the past under the same conditions.

Take, for example, coin flips in craps gambling. Every coin has two sides, with two different probabilities attached to either coming up when flipped (Heads or Tails).

This means there are four potential outcomes to any given coin flip. Two will be right-side up, and the other two will be the wrong side up. The probability of either outcome is 50% because there are equal chances that it could either happen.

Further, in the case of a die roll, which has six sides with various probabilities attached to each number showing on top after being rolled, it adds up to 100% because each possible outcome would occur one time out of six possibilities.

Empirical probabilities are used when experimental errors have been found in previous encounters involving similar events.

Once found, these discrepancies are studied until their reasons are known and corrected so future recordings match what should have happened based on the estimated probabilities.

For example, if a coin flip has come up tails four times out of ten flips, the empirical probability of tails coming up would be 40%.

The chance would only be 50% that it will land on heads or tails because based on past recurrences and how often each side appeared (or didn’t appear), there should only be a 50/50 chance for either outcome to occur.

Basic of Empirical Probability

Empirical probability is a method of estimation based on values that are already known to us. It’s the opposite of theoretical probability, which uses an equation to predict future events.

This can be used in real-life situations like guessing if your student loan will default or how likely it is for you to survive through childhood (given your current age).

How Do We Use Empirical Probabilities?

Well, first, we take our best guess for how likely an event might happen, and then we test it empirically, meaning by using real data. For example, let’s say that I want to know my chances of surviving into adulthood (we’ll pretend that this is possible given my size and weight).

Well, first, I’d need to take my best guess at how likely I am to survive childhood. Let’s say that if you’re in America, your chance of survival is about 98%. The next step is taking that number and testing it empirically.

So if we’re using the U.S. Social Security Administration records (they also have an excellent website) for this example, let’s check out Baby Boy Milgrom, born in 1992 in New York City, NY (also known as The Big Apple).

I bet you weren’t expecting me to go through all of those steps! But since we did our empirical probability at the end there, I think it’s pretty accurate (that website is fantastic). You can see that his chance of survival was about 99.5%. That’s pretty close to the 98% theoretical chance that I had guessed at all.

There is something I didn’t cover in that example, like how my method is suitable for comparing two distinct probabilities (like if you want to compare my chances of surviving childhood with your chances).

Still, it’s not so great to look at trends in data or find out how likely an event might be over time. Still, though, the empirical probability is a cool way to look at old statistics and find new ways to use them.

The Formula for Empirical Probability:

Probability of event = number of times the event happened/ total number of trials * 100%

For our example, let’s say that I performed this experiment one hundred times.

Event = surviving childhood

Number of times event happened = 15

The total number of trials=100, since each trial is an experiment.

The empirical probability for my survival turns out to be 15/100*100%=15%. We can round down, so the empirical probability is about 15%. This means that my chances are slightly less than 1 in 6. Pretty slim odds but at least not too bad.

Why Do We Need Empirical Probability?

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Empirical probabilities are a great way to think about the world around us. Using real-life data, we can learn new things and make informed decisions about our everyday lives.

Empirical probability is a handy tool used by statisticians and mathematicians to calculate numerical probabilities based on historical data.

An empirical probability is calculated by looking at the number of times an event has happened and dividing it by the total number of trials (or experiments).

Empirical probability is also an excellent way to see how much of an advantage you might have over someone else.

For example, you might know that your chances of surviving childhood are 60% higher than those of your best friend from Bosnia, but then you might find out that kids born in Bosnia only have a 40% chance of survival while their counterparts born in Sweden have 99%.

In this case, being Swedish is going to give you a pretty big leg up! Because empirical probability is so easy to do, there are many different empirical probabilities that you can find by just searching the world wide web.

For example, did you know that since 2006, Americans have had about a 7 out of 10 chance of seeing at least one sunrise every day?

For example, I was born in 1992 and thus had an empirical probability of 98% chance of surviving childhood, considering that about 95

Advantages of Empirical Probability

  • Empirical probability is a great tool to use in the real world.
  • Another advantage is that Empirical Probability is a handy tool used by statisticians and mathematicians.
  • Empirical probability allows you to make intelligent decisions based on data. In my example, I found out that my chances of survival were pretty slim, but I also learned a lot about how people in Bosnia survive childhood and the value of growing up in Sweden.
  • Empirical probability can be a great way to learn about trends over time or how most people live. Many websites have historical statistics that they’ve gathered over decades. You can use them to extrapolate things like values for different countries or certain quirks about populations (i.e., it’s probably not good if your friends keep dying).
  • Empirical probability allows us to make intelligent decisions about everyday life. Empirical probability is an extremely useful tool used by statisticians and mathematicians to calculate numerical probabilities based on historical data.
  • Empirical probability is a great way to learn about trends over time or how most people live. Many websites have historical statistics that they’ve gathered over decades and can extrapolate things like values for different countries or certain quirks about populations.
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Disadvantages of Empirical Probability

  • Empirical probability is not so good for trend analysis and prediction.
  • Historical information isn’t easy to find all the time. Sometimes you’ll have the link to the source but not the data itself (crucial for an empirical probability). Such an example might be like this: “Estimates show that there are about 1,000,000 dogs in Brazil.” You can use Google to search for these figures (that’s why people call it a ‘research engine’), but you’re still left with only estimates and no details on how they were found. They could’ve done an anonymous survey or just guessed based on looking around one neighborhood at a time. There’s also no way to know if their data was good enough for each dog to be truly counted. It may seem silly, but those details are essential for an academic article.
  • Another disadvantage is that outliers may skew the results.
  • Empirical probability is not so great for comparing two different probabilities (for example, I can’t empirically calculate the difference in your chances of survival versus mine)
  • Empirical probability is also not good for making specific predictions (i.e., what’s your chance of survival if you’re a girl born to Swedish parents living in Bosnia). It doesn’t give you hard numbers like p(surviving) = 0.6; rather, it usually gives you ranges (i.e., “your chances of survival are 60% higher than those of your best friend from Bosnia”).
  • Comparisons between groups can be difficult since empirical probability numbers are given for each group (as opposed to comparisons between data sets with one number).
  • Empirical probabilities usually only give you the probability of something happening at least once. This can mislead you into assuming that there’s no chance it could happen more than once (i.e., if I asked people how many times they’d seen a sunrise in their life and then averaged the responses together, it would be shallow since the average is skewed up by people who have never seen one before). This also means that your results can’t tell you about another group (since they’ve already got their own set of numbers); instead, what you get tells you only about yourself.
  • Empirical probability can be hard to calculate if you don’t know how many trials (experiments) there were
  • Empirical probability is not highly accurate, but it’s good enough for practical purposes.
  • Empirical probability can’t give you information on rare things. For example, the chances of being struck by lightning are estimated to be 1 in 3 million, so if your chance of surviving childhood were only 40%, it probably wouldn’t be a good idea to go outside during a thunderstorm.
  • Empirical probability can have some accuracy issues since scientists sometimes extrapolate data from incomplete sets to reach their conclusions instead of waiting for the experiments to finish.

However, these disadvantages are easily outweighed by the advantages. Empirical probability is an extremely useful tool used by statisticians and mathematicians alike to make informed decisions about our everyday lives and learn new things based on real-life data.

Although there can be some accuracy issues with empirical probability depending on how you interpret the data, it’s still a great way to learn more about trends over time or how most people live their lives.

This allows us to make sound and intelligent decisions based on data, which we need to progress as a species (or at least survive).

Not only does this improve our understanding of life around us, but it also makes us feel special to know that we’re not the only ones who get to see their sunrises every day.

Thus, the empirical probability is a handy tool used by statisticians and mathematicians alike to make informed decisions about our everyday lives and learn new things based on real-life data.

Although there can be some accuracy issues depending on how you interpret the data, it’s still a great way to learn more about trends over time or how most people live their lives.

This allows us to make intelligent decisions based on data, which we need to progress as a species (or at least survive). Not only does this improve our understanding of life around us, but it also makes us feel special to know that we’re not the only ones who get to see their sunrises every day.

Is Empirical Probability Reliable?

Mathematical models are powerful tools that have helped humans understand their world since ancient civilisations first discovered some of the core concepts behind modern mathematics.

However, many people believe that models are just approximations of reality at best and lies at worst. After all, how could something as complex as reality be contained within rules created by human beings?

The answer lies in probability, which is the concept that if something has an infinite number of outcomes, all possible outcomes are inherently likely.

What About Statistics?

When most people use the term “Probability,” they refer to statistical probability, which is very different from empirical probability. Statistical probabilities are numerical estimates about things that will happen based on mathematical models of what has already happened in the past.

For example, if I were trying to calculate how many smartphones Apple will sell next year, a statistical probability would be appropriate because it will happen in the future.

However, keep in mind that even though statistically-based models are usually more accurate than empirical ones, there is no way to tell whether they are correct until after the fact. This makes statistical probabilities useless when making essential decisions where failure could have serious consequences.

For these kinds of decisions, people should only use empirical probabilities as their guide because they are the only ones that can be tested in real-time by observing what’s happening around us.

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It doesn’t matter whether or not an experiment is biased as long as it can predict the outcome we’re looking for. Of course, there will always be certain things we cannot know without making a numerical estimate about them, but this shouldn’t stop us from discovering empirical probabilities whenever we can.

After all, the more specific and accurate our assumptions and predictions about how something works, the better chance we can influence its behavior.

Different Types of Statistical Model

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There are three types of probabilities that are commonly used in the real world today;

  • Numerical probability can be seen as an estimation of something’s likelihood relative to other possible outcomes. This is best exemplified by flipping a coin. If you flip a coin twice, the probability of getting heads both times is 0.125 or 12.5%, meaning that event is unlikely but not impossible. However, this only works if we assume that the results are random and that each flip has an equal chance of coming up heads or tails.
  • Empirical probability: which refers to the actual outcomes from a particular experiment based on real-world data. For example, suppose I wanted to know what percentage of people have brown eyes in America. In that case, I could take a sample size from random parts of the population and try to come up with an exact percentage by counting how many people have brown eyes blue eyes vs. other eye colors. Empirical probability is extremely useful in the scientific field because it allows us to make informed decisions based on data rather than hunches or assumptions.
  • Theoretical probability is the likelihood that an event will happen within a specific interval, given certain conditions. Theoretical probability can be seen as an attempt to predict what will happen in results based on mathematical models and statistics. However, this only works if all possible outcomes are considered equally likely (which isn’t always true). Furthermore, since we’re not dealing with the real world here but instead using numbers and symbols to represent some abstract concept, some people feel that making assumptions and predictions about concepts outside of our reality is useless at best and dangerous at worst (imagine if we used the wrong mathematical model to predict the weather and everyone died as a result).

Empirical Probability vs. Theoretical Probability

Numerical probability is not always the best tool to use in statistical models. While it does help us make educated guesses about what might happen, there are some situations where numerical probability cannot be used, and empirical probabilities should be considered instead.

For example, say we want to know what percentage of people have brown eyes in America. It would take someone hundreds of hours at least to count how many blue-eyed people live here versus those with brown or other eye colors.

This makes it impossible for anyone but a wealthy person or large company to gather enough data for an accurate percentage.

This is where empirical probability comes in: If we take a sample of people from America and count how many of them have brown eyes, we can estimate the entire population based on that data alone. However, this only functions if certain conditions are met before it’s conducted.

For example, our sample size must be large enough to reliably represent everyone else who lives here (which means at least several hundred people).

It must also be done randomly so that no group or demographic has more people represented than others. Furthermore, what matters most is determining whether or not any biases are being introduced into the results by the process used for experimenting itself.

As long as these conditions are met, empirical probabilities provide us with a highly cost-efficient way of testing many variables at once. For example, suppose I wanted to know what percentage of teenagers own a smartphone but didn’t have enough time or money to gather the required data myself.

In that case, I could send surveys to random high school students across the country and see how many respond with the correct answer.

However, empirical probabilities are best used in cases where there is no other option. Numerical probability may be our only choice if we need accurate predictions about future events (like who will win an election or how much it will rain next week). This is because numerical probability can consider all possible outcomes, unlike empirical probabilities, which are often biased by human error, poor sample size, or other factors.

Empirical Probability in Everyday Life

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Empirical probabilities are everywhere in our everyday lives. For example, if you were to flip a coin 100 times and it came up heads about 55 times, you could reasonably conclude that the next time it’s reversed the same number of times, there’s an over 50% chance it will come up tails (1).

Of course, this is only true IF all other things remain equal (the original coin doesn’t get bent in half somehow, for example).

But since we can’t see into the future, empirical probabilities are better than nothing when trying to make educated guesses about what might happen when observing real-world phenomena.

Key Note

Mathematical models are powerful tools that have helped humans understand their world since ancient civilisations first discovered some of the core concepts behind modern mathematics.

However, many people believe that models are just approximations of reality at best and lies at worst. After all, how could something as complex as reality be contained within rules created by human beings?

The answer lies in probability, which is the concept that if something has an infinite number of outcomes, all possible outcomes are inherently likely.

There are still things we cannot know for sure, but observing the real world is the only way to find out what those things are. After all, a model can never show us anything that isn’t there in the first place.

Empirical probability is essential because it’s based on observation which gives us results that may be biased by human error or poor sample size, but this doesn’t matter as long as it can predict whatever outcome we’re looking for.

Suppose we want to make better decisions about the future. In that case, our best bet is to use empirical probabilities instead of statistical probabilities because they can be tested and used right away.

Mathematical models have been around since ancient times and have helped humans understand their surroundings ever since, but always keep in mind that they’re only tools and not the actual thing we’re trying to measure.