Unit 12 · 2.8

Binomial distribution

binomialpdf for exactly; binomialcdf lower/upper for ranges

Locked

Pass Unit 11 (Expected value) with 4 of 5 on its checkpoint to unlock this unit. Exam is tomorrow, so you can also open it now.

Rule cards

Binomial when: 2 outcomes · fixed n trials · independent (with replacement or a %) · same p each time. P(X) = C(n, X) · pˣ · (1 − p)ⁿ⁻ˣ.

P(X) = C(n, X) pˣ (1 − p)ⁿ⁻ˣ (P(X = x) = ₙCₓ pˣ qⁿ⁻ˣ)

Unusual if P < 0.05.
Guessing 20 questions, 5 choices (p = .2), at least 15 right = 1.8E−7 ≈ 0 → unlikely.

Traps

Used the failure probability as p.
brown eyes 85% → p = .15
✓ p = probability of the thing you count = .85
Left out C(n, X).
exactly 1 of 6 → (.85)(.15)⁵
✓ C(6,1)(.85)(.15)⁵ = .000387
Left out (1 − p)ⁿ⁻ˣ.
C(6,1)(.85)
✓ × (.15)⁵

TI-84 Evo keys

Translate the words into the lower and upper values before typing anything; 'between a and b' includes both ends.

exactly k → binomialpdf(n, p, k)
at most k (≤ k) → cdf lower 0, upper k
less than k (< k) → cdf lower 0, upper k − 1
at least k (≥ k) → cdf lower k, upper n
more than k (> k) → cdf lower k + 1, upper n
between a and b → cdf lower a, upper b (both ends in): 33 to 35 → .155 (exam Q30)

Traps

Included or excluded an endpoint wrongly.
more than 5 → lower 5
✓ more than 5 → lower 6, upper n; less than 5 → lower 0, upper 4
Computed the opposite tail.
at least 25 of 50 → lower 0, upper 24 = .076
✓ lower 25, upper 50 = .924 (exam Q29)
Used pdf for a range.
at least 25 → binomialpdf(50,.59,25)
✓ a range always means cdf
Used cdf for 'exactly'.
exactly 30 → cdf 0 to 30
✓ exactly → binomialpdf = .113

TI-84 Evo keys