Binomial Probability
Learn to calculate probabilities when an experiment has exactly two outcomes and is repeated multiple times.
Definition
- = total number of trials
- = number of successes we want
- = probability of success on each trial
- = number of ways to choose successes from trials (combinations)
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Worked Examples
What is the probability of getting exactly 3 heads in 5 coin flips?
Identify the parameters
flips, heads, (fair coin) → , ,
Calculate the combination
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Calculate
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Calculate
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Multiply all parts
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Answer: The probability of getting exactly 3 heads in 5 flips is or
Common Mistakes
Forgetting to include the combination coefficient
Why it's wrong: The combination counts all the different ways to arrange successes among the trials. Without it, you only calculate one specific arrangement.
Correct: Always start with the combination:
Using instead of
Why it's wrong: The exponent represents the number of successes, not the total number of trials.
Correct: Success probability is raised to the power of successes: , and failure probability to the power of failures:
Applying binomial formula when trials are not independent
Why it's wrong: Binomial probability requires that each trial's outcome doesn't affect others. Drawing cards without replacement, for example, changes probabilities.
Correct: Check independence: Does knowing one result change the probability of another? If yes, binomial doesn't apply.
Interactive Visual
Coin Flipper
Click the button to run trials and see probability in action!
Bar Chart
Interactive Sandbox
Expression Calculator
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History
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Practice Problems
18 problemsWhich of the following is a requirement for a binomial experiment?
Why It Matters
- Medicine: Probability that exactly 7 out of 10 patients respond to a treatment
- Quality Control: Chance of finding 3 defective items in a batch of 20
- Sports: Likelihood of a basketball player making 8 out of 10 free throws
- Surveys: Probability that 60% of respondents choose a particular option
Real World Applications
Medical Trials
Researchers use binomial probability to evaluate treatment effectiveness.
Example:
If a drug has a 70% success rate, we can calculate the probability that exactly 8 out of 10 patients will respond positively.
A vaccine is 80% effective. In a group of 5 people who received it, what's the probability that exactly 4 are protected?
Calculate where ,
Step 1: Write the mathematical expression
Use:
Sports Analytics
Coaches use binomial probability to predict game outcomes and player performance.
Example:
A basketball player has a 75% free throw success rate. Binomial probability helps calculate the chance of making 9 out of 10 free throws.
A soccer player scores on 60% of penalty kicks. What's the probability of scoring exactly 3 out of 5 penalties?
Calculate where ,
Step 1: Write the mathematical expression
Use:
Key Takeaways
- 1Binomial probability applies when there are exactly two outcomes (success/failure) with fixed probability
- 2The formula is
- 3The combination counts all possible arrangements of successes
- 4Trials must be independent for binomial probability to apply
- 5Always check: fixed , constant , independent trials, two outcomes
Frequently Asked Questions
Glossary
- Binomial experiment
- An experiment with a fixed number of independent trials, each with exactly two outcomes (success/failure) and constant probability
- Trial
- A single repetition of the experiment (e.g., one coin flip, one question answered)
- Success
- The outcome we're counting (doesn't have to be 'good' - just the one we're interested in)
- Combination
- The number of ways to choose items from items, written as or
- Independent trials
- Trials where the outcome of one doesn't affect the probability of others
Formula Card
Binomial Probability Formula
$n$ = trials, $k$ = successes, $p$ = success probability, $\binom{n}{k}$ = combinations
Combination Formula
Number of ways to choose $k$ items from $n$ items