Binomial probability
Check the assumptions before counting successes across repeated trials.
Before you begin
- Independent events
- Combinations and factorials
Identify a binomial situation
A binomial model counts successes in a fixed number n of trials. Each trial has two categories, success and failure, the same success probability p, and independence between trials. The words success and failure are labels, not value judgments. Sampling without replacement from a small population usually breaks the constant-probability assumption.
Count the possible orders
One specific sequence with r successes and n − r failures has probability pʳ(1 − p)ⁿ⁻ʳ. There are n choose r arrangements with that count. Multiply by this combination count to find the probability of exactly r successes. 'At least one' is often easier to calculate as one minus the probability of zero.
Worked example
A fair coin is tossed independently four times. Find the probability of exactly two heads.
Show the worked solution
- There are C(4,2) = 6 arrangements of two heads.
- Each four-toss sequence has probability (1/2)⁴ = 1/16.
- Multiply the number of arrangements by their individual probability.
6/16 = 3/8.
Try it yourself
Which situation fits a binomial model?
Common mistakes
- Do not omit the combination factor for 'exactly r'.
What can you explain now?
Three independent trials each have success probability 0.2. Find the probability of at least one success.
Compare with the explanation
0.488.
The probability of no success is 0.8³ = 0.512. Its complement is 1 − 0.512 = 0.488.
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Common Core mathematics standards ↗