Question

The probability that an electronic component will fail in performance is 0.14.

Use the normal approximation to Binomial distribution to find the probability that among

100 such components,

(a) at most 12 will fail in performance.

(b) Let X be the number of components that fail. Find P(11 < X $\leq$ 16).

Answer 1

The number of components which fail outof 100 randomly selected ones, is modelled here as:

$X \sim Bin(n = 100, p= 0.14)$

This is approximated by a normal distribution as:

$X \sim N(\mu = np, \sigma^2= np(1-p ) )$

ΧN N(μ = 100 % 0.14, σ? = 100 % 0.14 % 0.86)

X ~ N = 14,0= 12.04)

a) The probability here is computed as:

P(X <= 12)

Applying the continuity correction, we have here:
P(X < 12.5)

Converting it to a standard normal variable, we have here:
$P(Z < \frac{12.5 - 14}{\sqrt{12.04}})$

Getting it from the standard normal tables, we have here:

Therefore 0.3328 is the required probability here.

b) The probability required here is:
P(11 <= X <= 16)

Applying the continuity correction, we have here:
P(10.5 < X < 16.5)

Converting it to a standard normal variable, we have here:

$P( \frac{10.5 - 14 }{\sqrt{12.04}} <Z < \frac{16.5 - 14}{\sqrt{12.04}})$

Getting it from the standard normal tables, we have here:

Therefore 0.6078 is the required probability here.