Question

Fifty-six percent of all american workers have a workplace retirement plan. if americans retirement plans are independent, what is the probability that exactly 3 out of 5 randomly selected americans will have a retirement plan

Answer 1

We have been given that 56 percent of all American workers have a workplace retirement plan and we are also given that  Americans retirement plans at workplace are independent.

To find out  probability of exactly 3 out of 5 randomly selected Americans will have a retirement plan we will use Bernoulli trials.

$_{r}^{n}\textrm{c}\cdot p^{r}\cdot q^{n-r}$

where p is probability of a success which in this case is Americans that have workplace retirement plans  and q is probability of failure which in this case are Americans that don't have workplace retirement plans.

Upon substituting our given values in this formula we will get,

$_{3}^{5}\textrm{c}\cdot0.56^{3}\cdot 0.44^{(5-3)}$

$_{3}^{5}\textrm{c}\cdot0.56^{3}\cdot 0.44^{2}$

$\frac{5!}{2!3!}\cdot0.175616\cdot 0.1936$

$\frac{5\cdot 4\cdot 3!}{2\cdot 1\cdot 3!}\cdot 0.56^{3}\cdot 0.44^{2}$

$10\cdot 0.175616\cdot 0.1936=0.339992576$

Rounding our answer to nearest hundredth we get our probability that exactly 3 out of 5 randomly selected Americans will have a retirement plan is 0.34.

Answer 2

The probability would be 0.34 or 34%.

We will set this up as a binomial distribution: