A binomial distribution question with given $p$ and $n$

Question

The question asks that it is known that $20\%$ of the computer chips produced by a manufacturer are defective. The Faculty of Arts & Science has just purchased $36$ new computers, each containing a chip produced by this manufacturer. Approximately what is the probability that at most $10$ of the computers contain a defective chip?

I've tried to solve this question by solving $$P(D \leq 10) = \sum_{x=0}^{10}{36 \choose x}(0.2)^x(0.8)^{36-x} = 0.911$$ which is not one of the answers below

(a) $0.9162$

(b) $0.3737$

(c) $0.7054$

(d) $0.02946$

(e) $0$

What did I do wrong?

Answer 1

Your answer is correct. The answer key is using the normal approximation to binomial with a continuity correction, then rounding to enter printed normal tables:

mu = 36*.2;  sg = sqrt(36*.2*.8)
z = (10.5 - mu)/sg;  z
[1] 1.375

Round up 1.38 to enter a printed normal CDF table and obtain Answer (a) 0.9162, which would have been an acceptable approximation--maybe 25 years ago.

round(pnorm(1.38), 4)
[1] 0.9162

Of course, the exact answer is obtained in R as 0.9111, to four places.

pbinom(10, 36, .2)
[1] 0.9110872

With $p = .2$ and $n = 36$ this binomial distribution (barely) meets some of the criteria for possible normal approximation, but $p = .2$ is pretty far from $1/2,$ so normal approximation doesn't work so well--as the following figure illustrates.

x = 0:36;  pdf = dbinom(x, 36, .2)
plot(x, pdf, type="h", lwd=2)
 abline(h=0, col="green2")
 abline(v=0, col="green2")
 abline(v=10.5, col="red", lwd=2, lty="dotted")
 curve(dnorm(x, mu, sg), add=T, col="blue")