# Definition and Formula of Binomial Distribution

• The binomial distribution is described mathematically by the binomial probability formula:
• P(X = r) = C(n, r)p^rq^(n-r)
• In the formula, P(X = r) indicates the probability of ‘r’ successes out of ‘n’ trials.

• C(n, r)’ denotes the number of combinations of ‘n’ items taken ‘r’ at a time.

• p’ is the probability of success on a single trial and ‘q’ is the probability of failure on a single trial (so q = 1-p).

# Binomial Probability Calculations

• The mean of a binomial distribution is given by μ = np, where ‘n’ is the number of trials and ‘p’ is the probability of success.

• The variance of a binomial distribution is σ^2 = npq, where ‘q’ is the probability of failure.

• The standard deviation is the square root of the variance, thus σ = √npq.

• To calculate binomial probabilities directly, use the binomial probability formula.

• To find cumulative probabilities, add up the individual probabilities.

# Using the Binomial Distribution Table

• Binomial distribution tables provide the cumulative probability of ‘r’ or fewer successes.

• If an upper value ‘r’ is given, subtract the cumulative probability from 1 to find the probability of more than ‘r’ successes.

• The table only provides values for the cumulative probabilities, so probabilities for an exact number of successes need to be calculated directly or found by subtracting relevant cumulative probabilities.

# Examples

• If we are interested in the probability of three heads in five coin tosses, this could be calculated as follows using the binomial formula: P(X=3) = C(5,3)(0.5)^3(0.5)^2.

• Suppose you wanted to know the probability of 7 or fewer heads in 10 coin tosses. This can be found by looking up the value in a binomial table or by adding the individual probabilities for 0 through 7 heads.

• Imagine a test has 12 multiple-choice questions, each with four options, and you want to know the probability of guessing exactly five correctly. This would be given by P(X=5) = C(12,5)(0.25)^5(0.75)^7.