Binomial Distributions Made Easy: A Practical Guide for Everyday Understanding

What Is a Binomial Distribution?

A binomial distribution is used when something can only have two possible outcomes for each attempt, like success or failure.

For example:

• Success: You catch a football pass.
• Failure: You drop the football.

The binomial distribution helps you figure out how likely it is to get a certain number of successes when you repeat the same task several times.

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When Should You Use a Binomial Distribution?

You use a binomial distribution when:

1. You have a fixed number of tries (called trials). For example, you throw a football 10 times.
2. Each trial has only two outcomes: success (catch the ball) or failure (drop the ball).
3. The chance of success is the same every time. For example, you have a 70% chance of catching the football each time.
4. Each trial is independent, meaning the result of one try doesn’t affect the next.

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Example 1: Fantasy Football Wide Receiver

Let’s say your Fantasy Football wide receiver is targeted 10 times in a game, and he catches the ball 70% of the time. You want to know the chances that he will catch exactly 7 passes out of 10 targets.

Problem:

How likely is it that your wide receiver catches exactly 7 passes out of 10 targets?

Solution:

To find this, you can use Excel’s BINOM.DIST function.

In Excel, use the formula:

=BINOM.DIST(7, 10, 0.7, FALSE)

• 7 is the number of catches (successes) you’re interested in.
• 10 is the number of passes (trials).
• 0.7 is the chance of success (70%).
• FALSE gives you the probability for exactly 7 catches (not cumulative).

The result will show that the probability is 27%. So, there’s about a 27% chance that your wide receiver will catch exactly 7 passes.

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Breaking It Down:

• Number of Attempts (Trials): In this example, it’s 10 football targets.
• Success or Failure: Each trial has two outcomes: either catch the ball (success) or drop the ball (failure).
• Chance of Success: Here, the receiver has a 70% chance of catching the ball.
• Number of Successes: You want to know the probability of exactly 7 catches.

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Example 2: Coin Flips

Now imagine you flip a coin 5 times, and you want to know how likely it is to get exactly 3 heads.

Problem:

What are the chances of getting exactly 3 heads in 5 flips of a fair coin?

Solution:

You can use Excel’s BINOM.DIST function again.

In Excel, use the formula:

=BINOM.DIST(3, 5, 0.5, FALSE)

• 3 is the number of heads you’re interested in.
• 5 is the number of flips (trials).
• 0.5 is the chance of success (50% for heads).
• FALSE gives you the probability for exactly 3 heads.

The result will show the probability is 31%. So, there’s about a 31% chance of getting exactly 3 heads in 5 flips.

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What Does a Binomial Distribution Tell You?

A binomial distribution helps you answer two key questions:

• What’s the most likely outcome? It shows what will happen most often. For example, if your wide receiver catches 70% of his passes, 7 catches out of 10 is the most likely outcome.
• What are the unlikely results? It shows how rare or unlikely certain results are. For example, it’s unlikely he’ll catch all 10 passes or drop every single one.

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Example 3: Fantasy Football Quarterback

Let’s say your Fantasy Football quarterback completes 65% of his passes. In a game, he throws 20 passes, and you want to know how likely it is that he’ll complete exactly 13 passes.

Problem:

What are the chances of completing exactly 13 passes out of 20 attempts?

Solution:

Use Excel’s BINOM.DIST function.

In Excel, use the formula:

=BINOM.DIST(13, 20, 0.65, FALSE)

• 13 is the number of completions you’re interested in.
• 20 is the number of passes (trials).
• 0.65 is the chance of success (65% completion rate).
• FALSE gives you the probability for exactly 13 completions.

The result shows that the probability is 18%. So, there’s an 18% chance your quarterback will complete exactly 13 passes.

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What Is a Random Variable?

A random variable is a way of representing the possible outcomes of an event in numbers. Random variables can be:

• Discrete: These have a countable number of outcomes. For example:
  • The number of catches in football.
  • The number of heads in a coin flip.
• Continuous: These can take any value in a range. For example:
  • The time it takes to complete a race.

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Types of Discrete Random Variables:

1. Number of Catches in Football: You can count how many passes your wide receiver catches.
2. Number of Heads in a Coin Flip: You can count how many times a coin lands on heads after several flips.

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Probability Distribution Function (PDF):

A PDF shows the probability of each possible outcome. For example:

• If you flip a coin, the chance of getting heads is 50%.
• If you roll a die, the chance of rolling any specific number (1 through 6) is 1/6.

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Conclusion:

By thinking of events as either successes or failures, binomial distributions provide a simple and practical way to predict outcomes. Whether you’re calculating how many passes your quarterback will complete or how many heads you’ll get when flipping a coin, binomial distributions allow you to make informed predictions with ease.

You can use Excel formulas like BINOM.DIST to quickly find the probability of specific outcomes. Now, even without any complex math, you have a simple tool to make better predictions in real life!