Question

1. in a large shipment of smartphone chargers, only 2% are found to be defective. a quality control manager randomly selects 25 chargers from the shipment for inspection. let x be the number of defective chargers in the sample. a. explain why x is binomial. identify the number of trials (n) and the probability of success (p) b. find the probability that exactly one out of the 25 chargers is defective. show your work.

Explanation:

Step1: Verify binomial conditions

1. Fixed trials (25 selections).
2. Two outcomes: defective (success) or not.
3. Constant $p=0.02$ (large shipment, so sampling without replacement approximates independent trials).
4. Independent trials (random selection).

Identify $n=25$, $p=0.02$.

Step2: Use binomial probability formula

Binomial formula: $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$
For $k=1$, $n=25$, $p=0.02$:
$\binom{25}{1} = \frac{25!}{1!(25-1)!}=25$
$p^1=0.02^1=0.02$
$(1-p)^{25-1}=0.98^{24}$

Step3: Calculate the probability

$P(X=1)=25 \times 0.02 \times 0.98^{24}$
First compute $0.98^{24} \approx 0.6157$
Then $25 \times 0.02 = 0.5$
$0.5 \times 0.6157 = 0.30785$

Answer:

a. $X$ is binomial because there are a fixed number of independent trials, two mutually exclusive outcomes, and a constant probability of "success" (finding a defective charger).
Number of trials $n=25$, probability of success $p=0.02$.
b. $\approx 0.308$