📊 Complete Binomial Distribution Explorer

Visualize and solve all binomial probability problems with interactive tools!

① Exercise 1: Basic Probability

Compute P(X=k) for the binomial distribution B(n,p) where:

(i) n=6, p=1/3, k=3

P(X=3) = C(6,3) × (1/3)³ × (2/3)³ = 20 × (1/27) × (8/27) = 160/729
Answer: ≈ 0.2195 (21.95%)

(ii) n=10, p=1/5, k=4

P(X=4) = C(10,4) × (1/5)⁴ × (4/5)⁶ = 210 × 0.0016 × 0.262144
Answer: ≈ 0.0881 (8.81%)

(iii) n=9, p=1/2, k=7

P(X=7) = C(9,7) × (1/2)⁷ × (1/2)² = 36 × (1/128) × (1/4)
Answer: ≈ 0.0703 (7.03%)

② Exercise 2: Target Practice

The probability that Mr.Q hits a target is 1/4. He tries 10 times. Find the probability that he hits the target:

(i) Exactly 4 times

(ii) At least one time

(i) P(X=4) = C(10,4) × (1/4)⁴ × (3/4)⁶ ≈ 0.1460 (14.60%)
(ii) P(X≥1) = 1 - P(X=0) = 1 - (3/4)¹⁰ ≈ 0.9437 (94.37%)

③ Exercise 3: Mean & Variance

Find the mean and variance of X for:

(i) 100 coin tosses (X = heads)

μ = n×p = 100×0.5 = 50
σ² = n×p×(1-p) = 100×0.5×0.5 = 25

(ii) 240 die rolls (X = fours)

μ = 240×(1/6) = 40
σ² = 240×(1/6)×(5/6) ≈ 33.33

④ Exercise 4: Component Test

The probability a component survives is 3/4. Find probability exactly 3 of 5 survive.

P(X=3) = C(5,3) × (3/4)³ × (1/4)² = 10 × (27/64) × (1/16) = 270/1024
Answer: ≈ 0.2637 (26.37%)

⑤ Exercise 5: Defective Items

Defective rate is 5%. Inspector picks 10 items. Find probability:

(i) At least one defective

(ii) Exactly two defective

(i) P(X≥1) = 1 - (0.95)¹⁰ ≈ 0.4013 (40.13%)
(ii) P(X=2) = C(10,2) × (0.05)² × (0.95)⁸ ≈ 0.0746 (7.46%)

⑥ Exercise 6: Light Bulbs

Probability a bulb lasts ≥600 hours is 0.9. Among 12 bulbs:

(i) Exactly 10 last ≥600h

P(X=10) = C(12,10) × (0.9)¹⁰ × (0.1)² ≈ 0.2301

(ii) ≥11 last ≥600h

P(X≥11) = P(X=11) + P(X=12) ≈ 0.3766 + 0.2824 ≈ 0.6590

(iii) ≥2 fail before 600h

P(Y≥2) = 1 - P(Y=0) - P(Y=1) ≈ 1 - 0.2824 - 0.3766 ≈ 0.3410

⑦ Exercise 7: Find Parameters

For binomial X with μ=6 and σ=2:

μ = n×p = 6, σ² = n×p×(1-p) = 4 ⇒ p = 1/3, n = 18
(i) PMF: P(X=k) = C(18,k) × (1/3)^k × (2/3)^(18-k)
(ii) P(X=3) ≈ 0.0278 (2.78%)
(iii) P(X≥2) = 1 - P(X=0) - P(X=1) ≈ 0.9984 (99.84%)

⑧ Exercise 8: Find Distribution

For X~B(n,p) with n=6 and 4P(X=4)=P(X=2):

4×C(6,4)p⁴(1-p)² = C(6,2)p²(1-p)⁴ ⇒ p = 1/3
Distribution: B(6, 1/3)
Mean = 6×(1/3) = 2
Standard Deviation = √(6×(1/3)×(2/3)) ≈ 1.1547

⑨ Exercise 9: Find Mean/Variance

For binomial with 5 trials, P(1)=0.4096, P(2)=0.2048:

P(1) = C(5,1)p(1-p)⁴ = 0.4096
P(2) = C(5,2)p²(1-p)³ = 0.2048
Solving gives p = 0.2
Mean = 5×0.2 = 1
Variance = 5×0.2×0.8 = 0.8
🏆 You've Completed All Exercises! 🏆