Binomial Distribution Formula

The binomial distribution is a core concept of a discrete probability distribution of probability theory and matters describing the number of successes in a predetermined range of several independent experiments following Bernoulli distributions. This distribution is widely used in many fields like biology, economics, engineering, and so on to model the events that result in either Success or Failure.

What is the Binomial Distribution Formula?

The binomial distribution formula calculates the probability of obtaining exactly 𝑘 successes in 𝑛 trials, where 𝑝 is the probability of success in each trial. The formula is given by:

P(X = k) = binom{n}÷{k} pk (1-p){n-k}

Where:

• (P(X = k) ) is the probability of (k) successes in (n) trials.
• (binom{n}÷{k}) is the binomial coefficient, which calculates the number of ways to choose (k) successes out of (n) trials and is defined as:

binom{n}{k} = {n!}÷{k!(n-k)!}

Here, (n!) (n factorial) is the product of all positive integers up to ( n ).

• (p) is the probability of success on a single trial
• ( (1-p)) is the probability of failure on a single trial.

How to Use the Binomial Distribution Formula? 

1. Define the Parameters: Specify the number of trials (n), the number of successes (k), and the probability of success (p).

2. Calculate the Binomial Coefficient: Use the formula for the binomial coefficient to compute (binom{n}÷{k} ) when the set of objects contains (n) elements and (k) elements should be selected, but the order of selection does not matter.

3. Compute the Probability: To get the probability of getting k exact value for the binomial random variable X, substitute n,k, and p in the binomial distribution formula.

Example Calculation

For instance, assume you are flipping a fair coin ten times, and you wish to find the probability of ending up with 4 heads.

1. Define the Parameters: 

The number of trials performed is defined as ( n = 10 )

The experimental probability is based on the given data, and one can also compute the theoretical probability in the same way as ( k = 4 ) (number of successes, i.e., heads)

Substitute n, k, and p into the binomial distribution formula to find P(X=k).

2. Calculate the Binomial Coefficient:

(10÷4)=10!4!÷(10−4)!=10!÷4!6!=210 Compute the Probability:

𝑃(𝑋=4)=(10÷4)(0.5)⁴(1−0.5)10-4=210×(0.5)⁴×(0.5)⁶=210×0.510=210×1÷1024=0.205

So, the probability of getting exactly 4 heads in 10 coin flips is approximately 0.205.

What is the Formula for the Standard Deviation of a Binomial Distribution?

The standard deviation of a binomial distribution measures the amount of variation or dispersion in a set of values. For a binomial distribution with parameters

n (number of trials) and

p (probability of success), the standard deviation

σ is calculated using the formula:

𝜎=𝑛𝑝(1−𝑝)

Explanation and Example

• Define the Parameters: The number of trials can be defined as the number of trials, which is often represented by the capital letter N. Probability of success can be defined as the possibility of success, which can be represented by the small letter p.
• Compute the Variance: The Variance, conventionally represented as (sigma²), is computed as (np(1-p)).
• Compute the Standard Deviation: Next, divide by the square root of the Variance.

Example Calculation

Continuing with the previous example of flipping a coin 10 times:

1. Define the Parameters:

n=10 (number of trials)

𝑝=0.5

p=0.5 (probability of success)

2. Compute the Variance:

𝜎2=𝑛𝑝(1−𝑝)=10×0.5×(1−0.5)=10×0.5×0.5=2.5

3. Compute the Standard Deviation:

𝜎=𝑛𝑝(1−𝑝) = 2.5≈1.58

σ= np(1−p)​ = 2.5​ ≈1.58

Thus, the standard deviation of the number of heads in 10 coin flips is approximately 1.58.

Conclusion

The binomial distribution is an ideal tool for modelling situations with two outcomes in consecutive trials. By analyzing the binomial distribution formula, you can obtain the probability of a certain number of successes, and the formula for the standard deviation will further help in observing the spread of those successes. Understanding these ideas improves the ability of educators, researchers, and practitioners to forecast and assess data containing binary results.

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