Binomial Distribution Calculator

Calculator

Parameters

Number of trials (n):

Probability of success (p):

Calculation Type:

Value (X):

Distribution Chart

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Binomial Distribution: Definition, Formula, and Examples

Binomial Distribution

Definition: The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials. A Bernoulli trial is an experiment with two possible outcomes: success or failure.

Formula:

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

Where:

$n$ is the number of trials
$k$ is the number of successes
$p$ is the probability of success on each trial
\inom{n}{k} is the binomial coefficient

Examples: Suppose you flip a fair coin

10

times

(n = 10, p = 0.5)

. Let's calculate various probabilities:

Probability of getting exactly 6 heads: $P(X = 6) = \binom{10}{6} (0.5)^6 (0.5)^{10-6} \approx 0.2051$
Probability of getting between 3 and 7 heads $(3 < X < 7)$ : $P(3 < X < 7) = P(X = 4) + P(X = 5) + P(X = 6) \approx 0.2051 + 0.2461 + 0.2051 = 0.6563$
Probability of getting less than 5 heads $(X < 5)$ : $P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) \approx 0.0010 + 0.0107 + 0.0439 + 0.1172 + 0.2051 = 0.3779$
Probability of getting more than 4 heads $(X > 4)$ : $P(X > 4) = 1 - P(X \leq 4) \approx 1 - 0.3779 = 0.6221$

Properties of Binomial Distribution

Mean: $E(X) = np$
Variance: $Var(X) = np(1-p)$
Standard Deviation: $SD(X) = \sqrt{np(1 - p)}$

How to Calculate Binomial Probabilities in R?

1# Set parameters
2n <- 10  # number of trials
3p <- 0.5 # probability of success
4
5# Calculate P(X = 6)
6prob_equal_6 <- dbinom(6, size = n, prob = p)
7print(paste("P(X = 6):", prob_equal_6))
8
9# Calculate P(X <= 4)
10prob_less_equal_4 <- pbinom(4, size = n, prob = p)
11print(paste("P(X <= 4):", prob_less_equal_4))
12
13# Calculate P(X > 7)
14prob_greater_7 <- 1 - pbinom(7, size = n, prob = p)
15print(paste("P(X > 7):", prob_greater_7))
16
17# Calculate P(3 < X < 8)
18prob_between_3_and_8 <- pbinom(7, size = n, prob = p) - pbinom(3, size = n, prob = p)
19print(paste("P(3 < X < 8):", prob_between_3_and_8))
20
21# Calculate mean and variance
22mean <- n * p
23variance <- n * p * (1 - p)
24print(paste("Mean:", mean))
25print(paste("Variance:", variance))

How to Calculate Binomial Probabilities in Python?

Python

1import scipy.stats as stats
2
3# Set parameters
4n = 10  # number of trials
5p = 0.5 # probability of success
6
7# Calculate P(X = 6)
8prob_equal_6 = stats.binom.pmf(6, n, p)
9print(f"P(X = 6): {prob_equal_6:.4f}")
10
11# Calculate P(X <= 4)
12prob_less_equal_4 = stats.binom.cdf(4, n, p)
13print(f"P(X <= 4): {prob_less_equal_4:.4f}")
14
15# Calculate P(X > 7)
16prob_greater_7 = 1 - stats.binom.cdf(7, n, p)
17print(f"P(X > 7): {prob_greater_7:.4f}")
18
19# Calculate P(3 < X < 8)
20prob_between_3_and_8 = stats.binom.cdf(7, n, p) - stats.binom.cdf(3, n, p)
21print(f"P(3 < X < 8): {prob_between_3_and_8:.4f}")
22
23# Calculate mean and variance
24mean = n * p
25variance = n * p * (1 - p)
26print(f"Mean: {mean}")
27print(f"Variance: {variance}")

Applications

The binomial distribution is widely used in various fields:

1. Quality Control in Manufacturing

In manufacturing, quality control often involves determining the probability of defective products within a batch.

Example: A factory with a 2% defect rate can use the binomial distribution to calculate the probability of finding exactly 3 defective bulbs in a sample of 100, which is 0.1823. This helps monitor product quality and detect unusually high defect rates, enabling timely corrective action.

2. Epidemiology for Modeling Disease Outbreaks

Epidemiologists use the binomial distribution to model the probability of disease contraction in a population over a specific period.

Example: If 10% of a population is susceptible to a disease, the binomial distribution can calculate the likelihood of exactly 5 people testing positive in a survey of 50 individuals.

Application: This aids in estimating disease spread and designing outbreak management strategies under various exposure scenarios.

3. Finance for Analyzing Stock Market Movements

In finance, the binomial distribution is used to model stock price movements, particularly in options pricing models.

Example: For a stock with a 60% daily chance of increasing in value, the binomial distribution can estimate the probability of the stock rising on exactly 3 out of 5 trading days.

Application: Traders and analysts use this to estimate potential stock movement outcomes, informing decisions on buying or selling options.

4. Political Science for Election Predictions

The binomial distribution helps predict election outcomes by modeling voter support for candidates.

Example: In a local election where 55% of voters favor Candidate A, the binomial distribution can calculate the probability of exactly 120 out of 200 surveyed voters supporting the candidate.

Application: This assists political analysts in making election predictions based on sample data, informing campaign strategies and media forecasts.

5. Biology for Genetic Studies

In genetics, the binomial distribution models inheritance patterns and the likelihood of traits appearing in offspring.

Example: If a gene has a 25% chance of being passed from a parent to a child, the binomial distribution can calculate the probability of exactly 2 out of 4 children inheriting the gene.

Application: This is crucial in genetic counseling, helping families understand the risks of passing on specific traits or conditions to their offspring.

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