Probability Theory/Limit Theorem

Question

Hi guys i have two questions i´m struggling with.

1)The mean duration of education for a population is 12 years and the standard deviation is 2 years. What is the maximum probability that a randomly selected individual will have had less than 9 or more than 15 years of education?

2)Limit Theorem: A firm receives 100 applications for a vacant job. Assume that the invitation to a job interview is i.i.d (Bernoulli). Further, imagine that the probability of being invited to an interview is 1/2. What is the probability that the number of applications accepted for interview is larger than 65?

Thanks in advance!

Answer 1

"1)The mean duration of education for a population is 12 years and the standard deviation is 2 years. What is the maximum probability that a randomly selected individual will have had less than 9 or more than 15 years of education?"

Look up the probabilities of X< (9- 12)/2 and X> (15-12)/2 in a table of the standard normal distribution and add them. Wikipedia has one here: https://en.wikipedia.org/wiki/Standard_normal_table.

Answer 2

1) Let $X$ be the number of years of education of the selected individual. "$X < 9$ or $X > 15$" is equivalent to $|X-12| > 3$. I think we may safely assume that the probability that the individual has exactly 9 or 15 years of education is zero, so $\Pr(|X-12| > 3) = \Pr(|x-12| \ge 3)$. By Chebyshev's Inequality, $$\Pr\left(|X-2)| \ge \frac{3}{2} \cdot 2\right) \le \left(\frac{2}{3}\right)^2$$ i.e. the maximum probability is $(2/3)^2$.

We could give a tighter bound if we knew the distribution of the number of years of education, for example if we knew it was a Normal distribution, but the problem statement does not specify the distribution, and in this case there does not seem to be any reason to assume a Normal distribution.