Have a test on this please help me!!!

Question

According to the bureau of labor statistics, about 10% of the civilian labor force (both men and women) were 20 to 24 years of age in 1996. Using this data and the formula for the binomial distribution, calculate the probabilities that a randomly selected group of six members of the civilian labor force in 1996

(a) exactly two were 20 to 24 years of age?

(b) fewer than two were 20 to 24 years of age?

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Answer 1

The binomial distribution formula is (_nC_x) • p^x • (1 - p)^{n - x}

In this case p = 0.1 (probability of being 20 to 24 years old), and 1 - p = 0.9.

The total number selected is n, which is 6, and the number of successes for part (a) is 2.

So, (₆C₂) • 0.1² • (0.9) ⁴ = 0.098415.

For part (b), we need fewer than 2 to be between 20 and 24, which means 0 or 1. So we calculate both and add them together.

(₆C₀) • 0.1⁰ • (0.9) ⁶ = 0.531441

(₆C₁) • 0.1¹ • (0.9) ⁵ = 0.354294

The sum of these two numbers is 0.885735.