In a binomial experiment, each trial has two possible outcomes: success and failure. If we denote the probability of success as p, the probability of failure can be defined as:
Probability of failure = 1 – p
Here’s a more detailed explanation:
- In a binomial setup, each trial is independent, meaning the outcome of one trial does not affect the others. You can think of tossing a coin — the outcome of one toss doesn’t change the outcome of the next.
- Since there are only two outcomes, the sum of the probabilities of success and failure must equal 1 (the total probability rule). Thus, we arrive at the formula:
- P(success) + P(failure) = 1
Using the representation of probability of success as p, we can substitute:
p + P(failure) = 1
If we rearrange this formula, we isolate the probability of failure:
P(failure) = 1 – p
To illustrate this with an example, let’s assume you are conducting an experiment where the probability of success (p) is 0.7. To find the probability of failure:
P(failure) = 1 – 0.7 = 0.3
This means there is a 30% chance of failure in each trial of the experiment. Understanding this relationship is critical for analyzing binomial distributions and making further statistical inferences.