Bayes' Theorem & Binomial Distribution explained through a friendly whiteboard conversation, examples, visuals, and practice for school students.
Step 1
Notes
Step 2
Formulae
Step 3
Examples
Step 4
Practice
Revise the definition, formulae, important results, and exam method first.
Use the diagram or interactive tool to see the same formula in action.
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The binomial distribution is used when an experiment has repeated trials with two possible outcomes.
These outcomes are often called success and failure, but the words only label the two cases.
It is used in probability, surveys, quality checks, tests, and decisioaking.
A binomial experiment has a fixed number of trials.
Each trial has two outcomes.
The probability of success stays the same for each trial.
The trials are independent, so one trial does not change another.
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The binomial formula combines number of arrangements with probability of each arrangement.
Meaning: A trial is one repeat of an experiment.
Definition: In tossing a coin 5 times, each toss is a trial.
The number of trials is usually written as n.
Meaning: These are labels for the two outcomes.
Definition: Success is the outcome being counted. Failure is the other outcome.
If counting heads, head is success and tail is failure.
Meaning: One trial does not affect another.
Definition: Trials are independent if earlier outcomes do not change later probabilities.
Coin tosses are usually treated as independent.
Meaning: Find the chance of one exact count.
Definition: Cr , where .
nCr counts positions of successes.
Easy
Question: A fair coin is tossed 3 times. What are n and p for heads?
Medium
Question: Find the probability of exactly 2 heads in 3 fair coin tosses.
Exam-level
Question: A test has 5 independent trualse questions. A student guesses. Find probability of exactly 4 correct.
Using binomial formula when p changes
Why it happens: The conditions are not checked.
Correct approach: Check fixed n, two outcomes, same p, and independent trials.
Forgetting nCr
Why it happens: Only one arrangement is counted.
Correct approach: Use nCr to count all positions of successes.
Using r instead of for failures
Why it happens: Success and failure counts are mixed.
Correct approach: Failures are .
What does success mean in binomial probability?
It means the outcome being counted.
Why do we use nCr?
It counts the different positions where successes can occur.
Can there be more than two outcomes?
Not for a binomial experiment. It must have two labelled outcomes.
Practise the same concept from this article. Your student profile may guide entitlement and follop recommendations, but it does not replace this concept.