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Binomial: thick coating on 15 chips

Short answer

Using binomial n=15, p=0.7: P(X≥12)=0.2969; P(X≤3)=0.0001; P(X=10)=0.2061.

During one stage in the manufacture of integrated circuit chips, a coating must be applied. If 70% of chips receive a thick enough coating, find the probability that among 15 chips (i) at least 12 will have thick enough coatings; (ii) at most 3 will have thick enough coatings; (iii) exactly 10 will have thick enough coatings.

Let X = number of chips with a thick enough coating. n = 15, p = 0.7, q = 0.3, so X ~ B(15, 0.7):

P(X = x) = C(15, x) (0.7)ˣ (0.3)¹⁵⁻ˣ

(i) At least 12

P(12) = 455 × 0.7¹² × 0.3³ = 0.17004
P(13) = 105 × 0.7¹³ × 0.3² = 0.09156
P(14) =  15 × 0.7¹⁴ × 0.3  = 0.03052
P(15) =   1 × 0.7¹⁵        = 0.00475

P(X ≥ 12) = 0.17004 + 0.09156 + 0.03052 + 0.00475 = 0.2969

(ii) At most 3

P(0) = 0.3¹⁵                = 0.0000000143
P(1) = 15 × 0.7 × 0.3¹⁴     = 0.0000005022
P(2) = 105 × 0.7² × 0.3¹³   = 0.0000082028
P(3) = 455 × 0.7³ × 0.3¹²   = 0.0000829393

P(X ≤ 3) = 0.0000917 ≈ 0.0001 (practically zero)

(iii) Exactly 10

P(X = 10) = C(15, 10) (0.7)¹⁰ (0.3)⁵ = 3003 × 0.0282475 × 0.00243 = 0.2061

Answer: (i) 0.2969, (ii) 0.0001, (iii) 0.2061

✓ Answer checked by Claude Sonnet 5.5

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