The big idea: A binomial experiment has n identical, independent trials. Each trial has two outcomes: success (prob p) or failure. X counts total successes. This is binomial distribution B(n, p).
Example: flip coin 10 times (n=10), count heads (X = number of heads). Each flip: p = 0.5 for heads.
Four conditions for binomial
All four must be TRUE
- Fixed n: number of trials is known beforehand.
- Independent: result of one trial does not affect others.
- Two outcomes: each trial is success or failure only.
- Constant p: probability of success is same every trial.
Is this binomial?
Scenario 1: Roll die 20 times, count how many sixes. Scenario 2: Keep rolling die until you get a 6.
Step by step
- Scenario 1: Fixed n=20, two outcomes (six or not), p=1/6 constant. YES, binomial B(20, 1/6).
- Scenario 2: No fixed n — you stop when you get the 6. NOT binomial (breaks condition 1).
Final answer
Scenario 1 is binomial. Scenario 2 is not.
Calculating P(X = r)
The meaning: C(n,r) = ways to arrange r successes in n trials. pr = prob of r successes. (1-p)n-r = prob of n-r failures.
Calculate a probability
B(10, 0.5): Find P(X = 3).
Step by step
- Use formula: P(X=3) = C(10,3) × 0.53 × 0.57
- C(10,3) = 10!/(3! × 7!) = 120
- P(X=3) = 120 × (0.5)10 = 120/1024 ≈ 0.117
Final answer
P(X = 3) ≈ 0.117 or 11.7%.
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Properties of binomial
Worked example
B(20, 0.3): Find mean and SD.
Step by step
- E(X) = 20 × 0.3 = 6
- Var(X) = 20 × 0.3 × 0.7 = 4.2
- SD(X) = sqrt(4.2) ≈ 2.05
Final answer
Mean = 6 trials will be successes (on average). SD ≈ 2.05.
Shape changes with p: If p = 0.5: symmetric bell. If p < 0.5: skewed right. If p > 0.5: skewed left. As n grows large, shape approaches normal curve.
When to use binomial
| Context | Is it binomial? | Why/why not |
|---|---|---|
| Survey: ask 100 people yes/no question | YES | n=100, two outcomes, constant p |
| Blood tests: how many positive out of 50? | YES | Fixed trials, two outcomes per test |
| Defects: inspect items until finding first defective | NO | No fixed n |
| Shoot 8 free throws, count makes | MAYBE | Usually yes if skill is consistent |
In exams: Identify: n (number of trials), p (probability success), X = number of successes. State B(n, p) clearly.