What is going on?
A binomial model counts successes across a fixed number of independent trials with the same success probability. Success can mean anything you define: a heads result, a correct answer, or an item passing a check. The combination term counts how many trial sequences produce k successes.
Work through it
- Choose the number of trials and the chance of success on each trial.
- Choose the target number of successes, from zero to the number of trials.
- Compare the exact probability with the cumulative probabilities shown beneath it.
Put numbers to the idea
For 10 fair coin flips, the chance of exactly 5 heads is C(10,5) × 0.5¹⁰ = 252 / 1024 ≈ 24.61%. A specific sequence of 5 heads and 5 tails has only a 1 / 1024 chance; there are 252 such sequences.
A little more clarity
What is the difference between exact and cumulative probability?
Exact means X equals k. At most adds probabilities from zero through k; at least adds from k through n.
Can success probability be zero or one?
Yes. At zero, all probability is at zero successes. At one, all probability is at n successes.
What are the mean and standard deviation?
The mean is np and the standard deviation is √[np(1 − p)]. They describe the center and spread of the distribution.