Calculations Relating to Binomial Distribution

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).

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.

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.

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.