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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)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

Where:

  • nn is the number of trials
  • kk is the number of successes
  • pp is the probability of success on each trial
  • \inom{n}{k} is the binomial coefficient
Examples: Suppose you flip a fair coin 1010 times (n=10,p=0.5)(n = 10, p = 0.5). Let's calculate various probabilities:
  1. Probability of getting exactly 6 heads:P(X=6)=(106)(0.5)6(0.5)10−6≈0.2051P(X = 6) = \binom{10}{6} (0.5)^6 (0.5)^{10-6} \approx 0.2051
  2. Probability of getting between 3 and 7 heads (3<X<7)(3 < X < 7):P(3<X<7)=P(X=4)+P(X=5)+P(X=6)≈0.2051+0.2461+0.2051=0.6563P(3 < X < 7) = P(X = 4) + P(X = 5) + P(X = 6) \approx 0.2051 + 0.2461 + 0.2051 = 0.6563
  3. Probability of getting less than 5 heads (X<5)(X < 5):P(X<5)=P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)≈0.0010+0.0107+0.0439+0.1172+0.2051=0.3779P(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
  4. Probability of getting more than 4 heads (X>4)(X > 4):P(X>4)=1−P(X≤4)≈1−0.3779=0.6221P(X > 4) = 1 - P(X \leq 4) \approx 1 - 0.3779 = 0.6221

Properties of Binomial Distribution

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

How to Calculate Binomial Probabilities in R?

R
# Set parameters
n <- 10  # number of trials
p <- 0.5 # probability of success

# Calculate P(X = 6)
prob_equal_6 <- dbinom(6, size = n, prob = p)
print(paste("P(X = 6):", prob_equal_6))

# Calculate P(X <= 4)
prob_less_equal_4 <- pbinom(4, size = n, prob = p)
print(paste("P(X <= 4):", prob_less_equal_4))

# Calculate P(X > 7)
prob_greater_7 <- 1 - pbinom(7, size = n, prob = p)
print(paste("P(X > 7):", prob_greater_7))

# Calculate P(3 < X < 8)
prob_between_3_and_8 <- pbinom(7, size = n, prob = p) - pbinom(3, size = n, prob = p)
print(paste("P(3 < X < 8):", prob_between_3_and_8))

# Calculate mean and variance
mean <- n * p
variance <- n * p * (1 - p)
print(paste("Mean:", mean))
print(paste("Variance:", variance))

How to Calculate Binomial Probabilities in Python?

Python
import scipy.stats as stats

# Set parameters
n = 10  # number of trials
p = 0.5 # probability of success

# Calculate P(X = 6)
prob_equal_6 = stats.binom.pmf(6, n, p)
print(f"P(X = 6): {prob_equal_6:.4f}")

# Calculate P(X <= 4)
prob_less_equal_4 = stats.binom.cdf(4, n, p)
print(f"P(X <= 4): {prob_less_equal_4:.4f}")

# Calculate P(X > 7)
prob_greater_7 = 1 - stats.binom.cdf(7, n, p)
print(f"P(X > 7): {prob_greater_7:.4f}")

# Calculate P(3 < X < 8)
prob_between_3_and_8 = stats.binom.cdf(7, n, p) - stats.binom.cdf(3, n, p)
print(f"P(3 < X < 8): {prob_between_3_and_8:.4f}")

# Calculate mean and variance
mean = n * p
variance = n * p * (1 - p)
print(f"Mean: {mean}")
print(f"Variance: {variance}")

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