Concepts: Bernoulli trials and the binomial distribution
Probability18 min readLesson 6 of 8
6.1 Repeated trials with two outcomes
Many experiments repeat the same act several times with only two outcomes each time, success or failure. A single trial of this kind is a Bernoulli trial, and repetitions count as Bernoulli trials when three conditions hold. Each trial has exactly two outcomes and one of them is named success. The trials are independent, so the result of one trial does not change the probabilities in another. And the probability of success is the same in every trial.
Tossing a coin repeatedly is the standard example, with head and tail as the outcomes. Drawing cards with replacement is another, because the pack returns to its original state after every draw. Drawing without replacement is not a set of Bernoulli trials, since the probabilities change after each draw. That distinction decides whether the binomial formula may be used at all, and examiners look for it explicitly.
6.2 The binomial probability formula
Let there be n Bernoulli trials with success probability p and failure probability q equal to one minus p, and let the random variable count successes. The probability that the count takes the value r has three parts. The binomial coefficient counts the arrangements of the successes and failures. The power of p gives the probability of the successes, and the power of q gives the probability of the failures. Because the trials are independent, the order of the successes does not change the probability of a particular arrangement, which is why a single product works with a counting factor in front. The result is the binomial distribution, and for one trial it reduces to the simple two-value distribution.
6.3 Mean and variance
The mean and variance of the binomial random variable are simple functions of the parameters: the mean is n times p, and the variance is n times p times q. These follow from adding the means and variances of the n independent single trials, each of which has mean p and variance p times q. For ten tosses of a fair coin the mean is 5 and the variance is 2.5, with a standard deviation of about 1.58.
6.4 Using the formula in questions
The formula answers four standard question types. Exactly r uses the formula directly. At least r adds the terms from r to n, and at most r adds the terms from 0 to r. At least one is almost always faster through the complement, which is one minus the probability that every trial fails. A question asking for at most three successes expects a sum of four terms, while a question about at least eight successes in ten trials is often quicker from the other end.
6.5 Reading the parameters from the words
The work in most binomial questions is to identify n and p from the description. A fixed number of independent repetitions gives n, the probability of the named outcome in one repetition gives p, and the random variable is the count of that outcome. Phrases such as with replacement, independently, and each time signal Bernoulli trials, while without replacement signals that the binomial model does not apply. Once n and p are written, the computation is substitution, and the mean and variance are immediate. A small sketch of the distribution, with the bar at r = np, shows whether the question is asking for a tail or for the centre of the distribution.
6.6 Worked example
One in five items produced by a machine is defective. In a sample of ten items, find the probability that exactly two are defective, and the probability that at least one is defective.
The items are independent with a fixed defect probability, so the count is binomial with n = 10 and p = 0.2. For exactly two defectives the binomial coefficient is 45, the success factor is 0.2 squared, which is 0.04, and the failure factor is 0.8 to the power 8, about 0.1678. Multiplying gives about 0.302. For at least one defective, the complement is all ten good, with probability 0.8 to the power 10, about 0.1074, so the answer is about 0.8926. The mean of the distribution is np = 2 and the variance is npq = 1.6.
6.7 A second worked check
A student guesses every answer on a ten-question true-or-false test, so each question is correct with probability one half. The number correct is binomial with ten trials and success probability one half. Exactly five correct has probability the binomial coefficient ten choose five divided by two to the power ten, which is 252/1024, about 0.246. The mean is five and the variance is 2.5, so a score far from five is evidence that the student is not guessing at random.