The probability of reliable operation of a structure when a load is applied is 0.9 to find the probability that out

The probability of reliable operation of a structure when a load is applied is 0.9 to find the probability that out of 150 structures tested independently of each other, more than 20 will fail; exactly 20 will fail.

1. Event X: Exactly 20 structures fail. By the local Laplace theorem, we get:

q = 0.9;
p = 1 – q = 0.1;
n = 150;
k = 20;
r = 1 / √ (npq) = 1 / √ (150 * 0.1 * 0.9) = 1 / √ (15 * 0.9) = 1 / √13.5 ≈ 0.2722;
x = (k – np) r = (20 – 150 * 0.1) r = 5 * 0.2722 ≈ 1.36;
P (X) = P (n, k) = r * φ (x) ≈ 0.2722 * φ (1.36) ≈ 0.2722 * 0.1582 ≈ 0.043.
φ (x) is a Gaussian function.
2. Event Y: more than 20 structures fail. By Laplace’s integral theorem, we get:

q = 0.9;
p = 1 – q = 0.1;
n = 150;
k1 = 21;
k2 = 150;
r = 1 / √ (npq) ≈ 0.2722;
x1 = (k1 – np) r = (21 – 150 * 0.1) r = 6 * 0.2722 ≈ 1.63;
x2 = (k2 – np) r = (150 – 150 * 0.1) r = 135 * 0.2722 ≈ 36.74;
P (Y) = P (n, k1, k2) = Φ (x2) – Φ (x1) ≈ Φ (36.74) – Φ (1.63) ≈ 0.5 – 0.4484 ≈ 0.0516.
Φ (x) is the Laplace function.
Answer: 1) 0.043; 2) 0.052.