Three components are randomly sampled, one at a time, from a large lot. As each component is selected, it is tested. If it passes the test, a success (S) occurs; if it fails the test, a failure (F) occurs. Assume that 80% of the components in the lot will succeed in passing the test. Let X represent the number of successes among the three sampled components.
What are the possible values for X? And There Probabilities ?
As with a lot of statistical questions, the phrase "from a large lot" implies that the population you are drawing samples from is sufficiently large such that sampling without replacement will not change the probability of a success, i.e. the distribution of the sum of individual Bernoulli trials, each with probability of success 0.8, is a Binomial distribution with the same success parameter and n equal to the number of samples taken. That is $$Y_i \sim Bernoulli(p = 0.8)$$ so that $$ Y_i = \begin{cases} 1, \;\ Pr(0.8) \\ 0, \;\ Pr(0.2) \end{cases} \\ $$ and $$ X = \sum_{i=1}^n Y_i \;\ \Rightarrow X \sim Binomial(n=3,p=0.8) $$ where $$ Pr(X = x) = \binom{3}{x}(0.8)^{x}(0.2)^{3-x} $$ Therefore $$Pr(X = 0) = (0.2)^{3} = 0.008$$ $$Pr(X = 1) = 3(0.8)(0.2)^{2} = 0.096$$ $$Pr(X = 2) = 3(0.8)^{2}(0.2) = 0.384$$ $$Pr(X = 3) = (0.8)^{3} = 0.512$$
