Exactly two
For X ~ Bin(4, 1/2), find P(X = 2).
- There are C(4,2) = 6 arrangements.
- Each arrangement has probability (1/2)⁴ = 1/16.
Answer: 3/8
Binomial probabilities
Check when a binomial model applies, then calculate the probability of a given number of successes.
Before you start
You’ll need Counting with restrictions. Follow a link if you want to revise it first.
When the binomial model applies
A binomial model needs a fixed number of independent trials, two outcomes per trial, and the same success probability each time.
For exactly k successes, combine the probability of one arrangement with the number of arrangements: C(n,k)pᵏ(1−p)ⁿ⁻ᵏ.
“Exactly”, “at least” and “at most” specify different events. A complement can shorten a tail calculation.
In four trials, two successes can occur in six different positions. The probability calculation must include all six arrangements.
Counting the possible arrangements
Exactly one with unequal probabilities
For X ~ Bin(3, 1/3), find P(X = 1).
- There are three possible positions for the success.
- Multiply 3 × (1/3) × (2/3)².
Answer: 4/9
Forgetting C(n,k) counts only one ordering of the outcomes.
Practice questions
The first link uses smaller values. Move on when you can answer without referring to the examples. The difficulty settings aren’t tied to exam grades.
Practice is filtered to your selected course. Change course or options when needed.
Check your understanding
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Would sampling without replacement from a small population satisfy the independent-trials assumption?