Statistics and Probability

The binomial distribution is a probability distribution that summarises the likelihood that a value will take one of two independent values under a given set of parameters or assumptions. Specifically, it models the number of successes in a fixed number of trials where each trial has the same probability of success. The mean (or expected value) and variance are key characteristics of this distribution.

The mean of a binomial distribution is given by the formula \( \mu = n \times p \), where \( n \) is the number of trials and \( p \) is the probability of success in each trial. The variance is given by the formula \( \sigma^2 = n \times p \times (1-p) \). These values help to understand the distribution's central tendency and the spread of the probability across the number of successes.

Key formulae for the binomial distribution:

• Mean: \( \mu = n \times p \)
• Variance: \( \sigma^2 = n \times p \times (1-p) \)

Example:

Consider a scenario where a student answers 10 true/false questions and guesses each answer. Assuming the probability of guessing correctly is 0.5, the mean and variance of the number of correct answers can be calculated as:

Mean: \( \mu = n \times p = 10 \times 0.5 = 5 \)

Variance: \( \sigma^2 = n \times p \times (1-p) = 10 \times 0.5 \times (1-0.5) = 2.5 \)

Thus, the student can expect, on average, to guess correctly 5 out of 10 times, with a variance of 2.5 in the number of correct guesses.

Note: In examinations, students should use technology to find binomial probabilities, as the formal proof of mean and variance is not required.

If you use a TI-Nspire GDC there are instructions useful for this topic.

This Binomial Distribution Video is from Revision Village and is aimed at students taking the IB Maths Standard level course.

This Bicen Maths video clip shows everything you need to memorise on this topic for A Level Statistics.