approximately 12.3% of american high school students drop out of school before graduation. assume the variable is binomial. choose 14 students entering high school at random. find these probabilities. round intermediate calculations and final answers to three decimal places.

If approximately 12.3% of American high school students drop out of school before graduation, assuming that the variable is binomial and choosing 14 students entering high school at random, the probability that all 14 stay in school and graduate is 0.160

As per the question statement, approximately 12.3% of American high school students drop out of school before graduation, and assuming that the variable is binomial, we are choosing 14 students entering high school at random.

We are required to calculate the probability that all 14 stay in school and graduate.

Given, Drop-out percentage = 12.3,

i.e., Drop-out Probability = (12.3/100) = 0.123

Therefore, Graduate Probability = (1 - 0.123) = 0.877

Let, the Drop-out and Graduate Probabilities be denoted by two random variables, "q" and "p" respectively. Then, (p = 0.877) and (q = 0.123).

Now, P (all 14 staying in school and graduating) = P(X = 14)

And, using binomial distribution, we can write that,

[tex]P(X = r) = [\frac{N!}{r!(N - r)!}*p^{r} *q^{N-r}][/tex]

Here, (r = 14), (X = 14), (p = 0.877) and (q = 0.123). Substituting these values in the above-mentioned formula, we get,

[tex]P(X = 14) = [\frac{14!}{14!(14 - 14)!}*0.877^{14} *0.123^{(14-14)}]\\Or, P(X = 14) = [\frac{14!}{14!0!}*0.877^{14} *0.123^{0}]\\[/tex]
Or, P(X = 14) = {(14!)/(14! * 1)} * 0.877¹⁴ * 1

Or, P(X = 14) = [(14!)/(14!)] * 0.877¹⁴

Or, P(X = 14) = 1 * 0.877¹⁴

Or, P(X = 14) = 0.159219
Or, (PX = 14) ≈ 0.160

Probability: It is a branch of Mathematics, that deals with the numerical descriptions concerning the extent to which an event is likely to occur, or how likely it is that a proposition is true, and is measured by the ratio of the favorable cases to the whole number of cases possible.

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