Answer to Question #107959 in Statistics and Probability for Emmanuel Daniels

Question

(a)State the characteristics of a binomial experiment.

(b)The Health Service claims that only 20% of young adults can pass its
physical fitness test. Suppose nine young adults are randomly selected and each is
given the fitness test.
i. If W is the random variable that denotes the number of the nine young adults
that pass the test, what is the name given to W?
ii. What is the expected number of young adults who pass the test?
iii. What is the probability that at least one of the young adults passes the test?

Answer 1

(a) 1. The experiment consists of a sequence of $n$ smaller experiments called trials, where $n$ is fixed in advance of the experiment.

2. Each trial can result in one of the same two possible outcomes (dichotomous trials), which we generically denote by success (S) and failure (F).

3. The trials are independent, so that the outcome on any particular trial does not influence the outcome on any other trial.

4. The probability of success P(S) is constant from trial to trial; we denote this probability by p.

i. W is the binomial random variable. $W\sim Bin(n, p)$

$$P(W=x)=\binom{n}{x}p^x(1-p)^{n-x}$$

ii. What is the expected number of young adults who pass the test? 

Given that $p=0.2, n=9$

$$E(W)=np=9(0.2)=1.8$$

iii. What is the probability that at least one of the young adults passes the test?

$$P(W\geq1)=1-P(W=0)=$$

$$=1-\binom{9}{0}(0.2)^0(1-0.2)^{9-0}=$$

$$1-1(1)(0.8)^9=0.865782272\approx0.8658$$