Binomial approximation A fair die is rolled 800 times. Find the probability to get a 6 at least 150 times.
My attempt:
The probability to get exactly 150 6's is:
$\binom{800}{150}\left(\frac16\right)^{150}\left(\frac56\right)^{800-150}$
The required probability is the sum of getting 150, 151 etc.
I understand we must use some approximation here, because we can't do these calculations one by one.
Can you please help me?
 A: Let Xi be Bernoulli with p=1/6. The CLT says that X, the sum of number of rolls of a 6, is approximately normal with mean 800/6 and variance 800/6(5/6). The needed probability is Pr(X>=150). Standardize and compute the probability. Pr(Z>(150-mu)/stddev). With continuity correction, Pr(X>=149.5). The continuity correction should give a closer estimate to the truth.
A: One can use a normal approximation to normal to find the probability that
For $X \sim \mathsf{Binom}(n = 800, p = 1/6),$ one can use a normal approximation to find $P(X \ge 150) = 1 - P(X\le 149) = 0.06420055,$ correct
to 2 (maybe 3) places. Or one can use software to get a more precise value. Computation in R, where pbinom is a binomial CDF:
1 - pbinom(149, 800, 1/6)
[1] 0.06420055

In R, the normal approximation gives the $P(X \ge 150) \approx 0.06255,$ as shown
below:
n = 800;  p = 1/6;
mu = n*p;  sg = sqrt(n*p*(1-p))
mu;  sg
[1] 133.3333
[1] 10.54093
1 - pnorm(149.5, mu, sg)
[1] 0.06255116

You can also standardize to get a normal approximation in terms of a
standard normal random variable $Z = \frac{X - \mu}{\sigma}.$ I will leave
the details of using a printed standard normal table to you. You will not
get exactly the value below because some rounding is involved in using
printed tables. [The use of 149.5 is called the continuity correction, which often gives a slightly closer approximation.]
1 - pnorm((149.5-mu)/sg)
[1] 0.06255116

In the figure below, the desired probability is the sum of the heights of the
blue bars to the right of the vertical dotted line and the approximate
probability is the area under the orange curve to the right of that line.

x = 100:180;  PDF = dbinom(x, 800, 1/6)
hdr = "PDF of BINOM(800, 1/6) with Density of NORM(133.33, 10.54)"
plot(x, PDF, type="h", lwd=2, col="blue", main=hdr)
 curve(dnorm(x, mu, sg), add=T, lwd=2, col="orange")
 abline(h = 0, col="green2")
 abline(v = 149.5, lty = "dotted", lwd=2)

