Binomial Distribution: Methods of Computation

Question

I have problems with questions regarding probability with little information given. Is it simpler than it really is? I'm not sure where to begin to start.

For example:

40% of adults receive an email for a magazine subscription over a 12 month period. Consider a random sample of 15 adults:

(1). What is the probability that none of the sample received an email for a magazine subscription?

(2). What is the probability that over half of the sample received an email for a magazine subscription?

Is $n = 15$ or is it 40% of 15, being $n = 6$? How do I start?

Answer 1

Here is a start for (1): Let $X$ be the number who get such an email. Then $X \sim Binom(15, .4)$ You seek $P(X = 0) = (.6)^{15}.$

In R statistical software:

 dbinom(0, 15, .4)  # 'dbinom' is PDF
 ## 0.000470185
 .6^15
 ## 0.000470185

For (2), you want $P(X \ge 7.5) = P(X \ge 8).$ You have not provided any context or shown any attempt, so I don't know whether you are supposed to compute this on a calculator, look it up in a printed binomial table, or do a normal approximation.

$$P(X \ge 8) = \sum_{x=8}^{15} {15 \choose x} (.4)^x (.6)^{15-x} = \,??.$$

Normal approximation: $\mu = np = 15(.4) = 6$ and $\sigma = np(1-p) = 1.897$ $$P(X > 7.5) = P\left(\frac{X-\mu}{\sigma} > \frac{7.5 - 6}{1.897}\right) \approx P(Z > 0.79) = \,??,$$ where $Z$ is standard normal.

 1 - pbinom(7, 15, .4)  # 'pbinom' is CDF
 ## 0.2131032

Here is a bar chart of the distribution. Values for 0 and beyond 12 are too small to show on the plot.

 x = 0:15;  pdf = dbinom(x, 15, .4)
 plot(x, pdf, ylim=c(0, max(pdf)), lwd=2, type="h", main="PDF of BINOM(15, .4)")
 abline(h=0, col="green2")

Answer 2

Binomial distribution

Here $p = 40\% = 0.4, n = 15.$