A binomial distribution arises from repeated Bernoulli trials where an experiment produces only two possible mutually exclusive outcomes: 'success' and 'failure'. In this topic, students explore how to identify binomial experiments, use the probability formula \(P(X = r) = {}^nC_r\, p^r q^{n-r}\), plot probability distribution graphs, determine the symmetry or skewness of distributions, and compute the mean (\(\mu\)), variance (\(\sigma^2\)), and standard deviation (\(\sigma\)).
An easy-to-read summary of the four key binomial conditions, the core probability formula, and the formulas for mean, variance, and standard deviation.
A Bernoulli trial is an experiment with exactly two mutually exclusive outcomes: 'success' (probability \(p\)) and 'failure' (probability \(q = 1 - p\)). When \(n\) independent and identical Bernoulli trials are performed, the experiment becomes a binomial experiment, written as \(X \sim B(n, p)\), where \(X\) denotes the number of successes.
A Bernoulli trial is an experiment that produces only two possible outcomes: 'success' and 'failure'. The probability of 'success' is \(p\), and the probability of 'failure' is \(q = 1 - p\), where \(0 < p < 1\). The discrete random variable \(X = \{0, 1\}\), where \(0\) represents 'failure' and \(1\) represents 'success'. An experiment with \(n = 1\) is a Bernoulli trial.
An experiment consisting of \(n\) similar Bernoulli trials is known as a binomial experiment. The discrete random variable \(X\) represents the number of successes \(r\) obtained from \(n\) similar Bernoulli trials (\(r \in \{0, 1, 2, \dots, n\}\)). Notation: \(X \sim B(n, p)\).
For \(X \sim B(n, p)\), the binomial probability function is: \[P(X = r) = {}^nC_r\, p^r q^{n-r}, \quad r = 0, 1, 2, \dots, n\] where \(n\) = total number of trials, \(r\) = number of successful outcomes, \(p\) = probability of success in a single trial, \(q = 1 - p\) = probability of failure in a single trial, and \({}^nC_r\) = number of combinations of choosing \(r\) successes out of \(n\) trials.
A distribution table lists each discrete value \(r\) alongside its calculated probability \(P(X = r)\). A distribution graph displays values of \(r\) on the horizontal axis and \(P(X = r)\) on the vertical axis (represented as discrete vertical lines or spikes).
Mean (Expected Value), \(\mu\): \(\mu = np\). Variance, \(\sigma^2\): \(\sigma^2 = npq\). Standard Deviation, \(\sigma\): \(\sigma = \sqrt{npq}\). Standard deviation is the square root of the variance because variance measures squared deviations from the mean.
In situations requiring the determination of a minimum sample size \(n\) satisfying an inequality such as \((1 - p)^n < k\): apply logarithms on both sides: \(n \log(1 - p) < \log k\). Since \(0 < (1 - p) < 1\), \(\log(1 - p)\) is a negative value. Dividing by \(\log(1 - p)\) reverses the inequality sign to \(n > \frac{\log k}{\log(1 - p)}\). Round up to the next integer to find the minimum value of \(n\).
Write the inequality in the form \((1 - p)^n < k\).
Apply logarithms on both sides: \(n \log(1 - p) < \log k\).
Divide by \(\log(1 - p)\), which is negative, and reverse the inequality sign.
Round up to the next integer to find the minimum value of \(n\).
Binomial probability formula
For \(r = 0, 1, 2, \dots, n\).
Mean, variance and standard deviation
Use these to describe the distribution.
Answer in your mind, then press "Check Answer".
A discrete random variable \(X\) follows a binomial distribution \(X \sim B(5, 0.4)\). Find \(P(X = 2)\).
A binomial random variable \(X \sim B(n, p)\) has a mean of \(45\) and a standard deviation of \(3\). Find the values of \(n\) and \(p\).
In a survey, \(54\%\) of Malaysians buy locally made cars. If \(8\) people who just bought new cars are selected at random, write down the formula expression to calculate the probability that at least \(2\) of them bought locally made cars.
Assuming every two-outcome experiment is binomial without checking independence: if items are selected without replacement from a small population, the probability changes from trial to trial, so it is not a binomial distribution.
A binomial experiment requires \(n\) independent trials with a constant probability of success. If the probability changes from trial to trial, it is not binomial.
Treating trials with more than two categories as binomial: experiments resulting in win, draw, or lose (3 outcomes) are not Bernoulli trials or binomial unless two categories are combined into a single "success" / "failure" definition.
A Bernoulli trial must have exactly two mutually exclusive outcomes. If there are more than two categories, they must be combined into two groups first.
Thinking the standard deviation is \(npq\): confusing variance (\(\sigma^2 = npq\)) with standard deviation (\(\sigma = \sqrt{npq}\)).
Variance is \(\sigma^2 = npq\), while standard deviation is the positive square root of variance: \(\sigma = \sqrt{npq}\).
Objective: Identify binomial experiments, calculate variance, and match key binomial distribution terms with their formulas or definitions.
1 Which of the following experiments represents a binomial experiment?
2 A discrete random variable \(X\) has a binomial distribution \(X \sim B(50, 0.2)\). What is the variance of \(X\)?
3 For a binomial distribution \(X \sim B(n, p)\), the mean \(\mu\) is given by the formula ________.
4 When the probability of success is \(p = 0.5\), the probability distribution graph of a binomial distribution is ________.
Drag each term to the correct formula or definition. On a phone, tap a term first, then tap the matching box.
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