## The Binomial Distribution - Statistics 1/2

Note: If you are looking for TI-Nspire notes on Binomial, click here.

There are four conditions that need to be met to use the binomial distribution, they are:

1 - There is a fixed number of trials n.

2 - There are only two possible outcomes - success and failure.

3 - The probability of success and failure remains the same for each trial.

4 – The n trials are independent.

The binomial distribution is written as:

X ~ B(n, p)

Where X is the variable, n is the number of trials, and p is the probability of success.

The mean for the binomial distribution E(X) = np

The variance for the binomial distribution Var(X) = npq

Where q is the probability of failure: q = 1 – p

P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ

P(X ≤ r) = Use the tables

P(X < r) = P(X ≤ r – 1) = Use the tables

P(X ≥ r) = 1 – P(X ≤ r – 1) = Use the tables

P(X > r) = 1 – P(X ≤ r) = Use the tables

P(4 < X < 9) = P(X ≤ 8) – P(X ≤ 4)

The binomial distribution can be approximated using the Poisson distribution if n is large (n > 50) and the probability of success p is very small (p < 0.2). If n is large and p is small then the mean np, and the variance npq will be very similar; for the Poisson distribution the mean and variance are same.

The binomial distribution can be approximated using the normal distribution if the number of trials n is very large (n > 50 generally), and the probability of success p is close to 0.5, so that the distribution is symmetrical. The normal distribution is considered accurate enough for the binomial distribution when np > 5, and nq > 5. The larger the value of np and nq then the more accurate the normal distribution will be.

When using the normal distribution to approximate a binomial distribution, you need to apply a continuity correction. The binomial distribution is for discrete variables, and the normal distribution is for continuous variables:

P(X < 15) = P(X < 14.5)

P(X > 12) = P(X > 12.5)

P(X ≤ 17) = P(X ≤ 17.5)

P(12 < X ≤ 15) = P(12.5 ≤ X ≤ 15.5)

For a continuous distribution such as the Normal Distribution:

P(X < 2) = P(X ≤ 2)

1 – Write the distribution: X ~ B(n, p)

2 – Write the probability you are trying to find:

P(X = 3)

P(X ≤ 4)

P(X > 7)

3 – Solve the probability by using the equation for a single value of X, or change the inequality sign to ≤ if you need to, and then use the binomial tables in the formula booklet.

P(X = 3) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ

P(X ≤ 4) = Use binomial tables

P(X > 7) = 1 – P(X ≤ 7) then use binomial tables

1 - There is a fixed number of trials n.

2 - There are only two possible outcomes - success and failure.

3 - The probability of success and failure remains the same for each trial.

4 – The n trials are independent.

The binomial distribution is written as:

X ~ B(n, p)

Where X is the variable, n is the number of trials, and p is the probability of success.

__Mean and Variance__The mean for the binomial distribution E(X) = np

The variance for the binomial distribution Var(X) = npq

Where q is the probability of failure: q = 1 – p

__Calculating probabilities__P(X = r) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ

P(X ≤ r) = Use the tables

P(X < r) = P(X ≤ r – 1) = Use the tables

P(X ≥ r) = 1 – P(X ≤ r – 1) = Use the tables

P(X > r) = 1 – P(X ≤ r) = Use the tables

P(4 < X < 9) = P(X ≤ 8) – P(X ≤ 4)

__Binomial Distribution being approximated by other distributions (Statistics 2)__The binomial distribution can be approximated using the Poisson distribution if n is large (n > 50) and the probability of success p is very small (p < 0.2). If n is large and p is small then the mean np, and the variance npq will be very similar; for the Poisson distribution the mean and variance are same.

The binomial distribution can be approximated using the normal distribution if the number of trials n is very large (n > 50 generally), and the probability of success p is close to 0.5, so that the distribution is symmetrical. The normal distribution is considered accurate enough for the binomial distribution when np > 5, and nq > 5. The larger the value of np and nq then the more accurate the normal distribution will be.

When using the normal distribution to approximate a binomial distribution, you need to apply a continuity correction. The binomial distribution is for discrete variables, and the normal distribution is for continuous variables:

P(X < 15) = P(X < 14.5)

P(X > 12) = P(X > 12.5)

P(X ≤ 17) = P(X ≤ 17.5)

P(12 < X ≤ 15) = P(12.5 ≤ X ≤ 15.5)

For a continuous distribution such as the Normal Distribution:

P(X < 2) = P(X ≤ 2)

__Steps for answering a Binomial Distribution Question__1 – Write the distribution: X ~ B(n, p)

2 – Write the probability you are trying to find:

P(X = 3)

P(X ≤ 4)

P(X > 7)

3 – Solve the probability by using the equation for a single value of X, or change the inequality sign to ≤ if you need to, and then use the binomial tables in the formula booklet.

P(X = 3) = ⁿCᵣ pʳ (1 – p)ⁿ⁻ʳ

P(X ≤ 4) = Use binomial tables

P(X > 7) = 1 – P(X ≤ 7) then use binomial tables