Understanding binomial experiments is the first step to understanding the binomial distribution.
This tutorial defines a binomial experiment and provides several examples of experiments that are and are not considered to be binomial experiments.
Binomial Experiment: Definition
A binomial experiment is an experiment that has the following four properties:
1. The experiment consists of n repeated trials. The number n can be any amount. For example, if we flip a coin 100 times, then n = 100.
2. Each trial has only two possible outcomes. We often call outcomes either a “success” or a “failure” but a “success” is just a label for something we’re counting. For example, when we flip a coin we might call a head a “success” and a tail a “failure.”
3. The probability of success, denoted p, is the same for each trial. In order for an experiment to be a true binomial experiment, the probability of “success” must be the same for each trial. For example, when we flip a coin, the probability of getting heads (“success”) is always the same each time we flip the coin.
4. Each trial is independent. This simply means that the outcome of one trial does not affect the outcome of another trial. For example, suppose we flip a coin and it lands on heads. The fact that it landed on heads doesn’t change the probability that it will land on heads on the next flip. Each flip (i.e. each “trial”) is independent.
Examples of Binomial Experiments
The following experiments are all examples of binomial experiments.
Example #1
Flip a coin 10 times. Record the number of times that it lands on tails.
This is a binomial experiment because it has the following four properties:
- The experiment consists of n repeated trials. In this case, there are 10 trials.
- Each trial has only two possible outcomes. The coin can only land on heads or tails.
- The probability of success is the same for each trial. If we define “success” as landing on heads, then the probability of success is exactly 0.5 for each trial.
- Each trial is independent. The outcome of one coin flip does not affect the outcome of any other coin flip.
Example #2
Roll a fair 6-sided die 20 times. Record the number of times that a 2 comes up.
This is a binomial experiment because it has the following four properties:
- The experiment consists of n repeated trials. In this case, there are 20 trials.
- Each trial has only two possible outcomes. If we define a 2 as a “success” then each time the die either lands on a 2 (a success) or some other number (a failure).
- The probability of success is the same for each trial. For each trial, the probability that the die lands on a 2 is 1/6. This probability does not change from one trial to the next.
- Each trial is independent. The outcome of one die roll does not affect the outcome of any other die roll.
Example #3
Tyler makes 70% of his free-throw attempts. Suppose he makes 15 attempts. Record the number of baskets he makes.
This is a binomial experiment because it has the following four properties:
- The experiment consists of n repeated trials. In this case, there are 15 trials.
- Each trial has only two possible outcomes. For each attempt, Tyler either makes the basket or misses it.
- The probability of success is the same for each trial. For each trial, the probability that Tyler makes the basket is 70%. This probability does not change from one trial to the next.
- Each trial is independent. The outcome of one free-throw attempt does not affect the outcome of any other free-throw attempt.
Examples that are not Binomial Experiments
Example #1
Ask 100 people how old they are.
This is not a binomial experiment because there are more than two possible outcomes.
Example #2
Roll a fair 6-sided die until a 5 comes up.
This is not a binomial experiment because there is not a pre-defined n number of trials. We have no idea how many rolls it will take until a 5 comes up.
Example #3
Pull 5 cards from a deck of cards.
This is not a binomial experiment because the outcome of one trial (e.g. pulling a certain card from the deck) affects the outcome of future trials.
A Binomial Experiment Example & Solution
The following example shows how to solve a question about a binomial experiment.
You flip a coin 10 times. What is the probability that the coin lands on heads exactly 7 times?
Whenever we’re interested in finding the probability of n successes in a binomial experiment, we must use the following formula:
P(exactly k successes) = nCk * pk * (1-p)n-k
where:
- n: the number of trials
- k: the number of successes
- C: the symbol for “combination”
- p: probability of success on a given trial
Plugging these numbers into the formula, we get:
P(7 heads) = 10C7 * 0.57 * (1-0.5)10-7 = (120) * (.0078125) * (.125) = 0.11719.
Thus, the probability that the coin lands on heads 7 times is 0.11719.