Binomial probability calculations
Worked example: P(X=r)
B(20, 0.3): Find P(X=5).
Solution
- Use formula or GDC (calculator)
- P(X=5)=C(20,5)×0.3⁵×0.7¹⁵
- Calculator gives ≈0.179
Final answer
P(X=5)≈17.9%. Use binomial function on calculator.
In exam: State n, p, r clearly.
Show formula.
Use GDC.
Write answer as decimal or percent.
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Cumulative probabilities
Notation: P(X≤r): probability of r or fewer successes.
P(X<r): fewer than r.
P(X≥r): r or more.
Worked example
B(20,0.3): Find P(X≤3).
Solution
- P(X≤3)=P(X=0)+P(X=1)+P(X=2)+P(X=3)
- Or use cumulative function on GDC
- P(X≤3)≈0.107
Final answer
P(X≤3)≈10.7%.
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Using complements
Strategy: P(X≥r) = 1 - P(X≤r-1).
Often easier to calculate complement.
Worked example
B(20,0.3): Find P(X>5).
Solution
- P(X>5)=1-P(X≤5)
- Find P(X≤5) using cumulative function
- P(X≤5)≈0.581
- P(X>5)=1-0.581=0.419
Final answer
P(X>5)≈41.9%. Complement easier than summing.
Bounded probabilities
P(r≤X≤s): Between two values: find P(X≤s) - P(X≤r-1).
Worked example
B(20,0.3): Find P(3≤X≤8).
Solution
- P(3≤X≤8)=P(X≤8)-P(X≤2)
- Use cumulative function twice
- ≈0.887-0.035=0.852
Final answer
P(3≤X≤8)≈85.2%.