The unemployment rate is 5.8% (Bureau of Labor Statistics, April 3, 2003). Suppose that 100 employable people are selected randomly.

1)What is the probability that exactly six people are unemployed (to 4 decimals)?

2)What is the probability that exactly six people are unemployed (to 4 decimals)?

3)What is the probability that at least four people are unemployed (to 4 decimals)?

An excel spreadsheet is very helpful

for these kinds of calculations.

You could calculate these pobabilities directly. Or, you could use a poisson distribution. I'll use a Poisson here.

the formula for the poisson is
P(k;l) = (l^k)*e^(-l) / k!
where l is the expected number = 5.8
and k is the observed number.

so for 1 (and 2, they are identical)
P(k;l) = 5.8^6 *e^(-5.8) / (1*2*3*4*5*6)
= .16

take it from here.

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To answer these questions, we can use the binomial distribution formula. The formula for finding the probability of exactly r successes in n trials is:

P(X = r) = (nCr) * p^r * (1 - p)^(n - r)

Where:
n = number of trials
r = number of successes
p = probability of success in a single trial
nCr = combination formula

In this case, we have:
n = 100 (number of employable people)
r = 6 (number of unemployed people)
p = 0.058 (unemployment rate)

1) To find the probability that exactly six people are unemployed, we can substitute the values into the formula:

P(X = 6) = (100C6) * (0.058^6) * (1 - 0.058)^(100 - 6)

Using a calculator or statistical software to evaluate this expression, we get a probability of approximately 0.1852.

2) The probability of exactly six people being unemployed is the same as in question 1. It is approximately 0.1852.

3) To find the probability that at least four people are unemployed, we need to calculate the probabilities for four, five, and six people being unemployed, and add them together.

P(X ≥ 4) = P(X = 4) + P(X = 5) + P(X = 6)

We can use the same formula as before, but need to sum up the probabilities for each value of r.

P(X = 4) = (100C4) * (0.058^4) * (1 - 0.058)^(100 - 4)

P(X = 5) = (100C5) * (0.058^5) * (1 - 0.058)^(100 - 5)

P(X = 6) = (100C6) * (0.058^6) * (1 - 0.058)^(100 - 6)

After calculating these probabilities separately, we can add them together to get the final result.

P(X ≥ 4) = P(X = 4) + P(X = 5) + P(X = 6)

Using a calculator or statistical software, we get a probability of approximately 0.2403.

To answer these questions, we need to use the binomial probability formula. The formula for the probability of exactly x successes in n independent Bernoulli trials is:

P(X = x) = (nCx) * p^x * (1-p)^(n-x)

Where:
- n is the number of trials (in this case, the number of employable people)
- x is the number of successes (in this case, the number of unemployed people)
- p is the probability of success (in this case, the unemployment rate)

Let's solve these questions step by step:

1) Probability of exactly six people being unemployed:
Using the formula, we have n = 100, x = 6, and p = 0.058 (unemployment rate of 5.8%).

P(X = 6) = (100C6) * 0.058^6 * (1-0.058)^(100-6)

To calculate this probability, we can use a calculator or software with statistical functions. The result is approximately 0.1673.

2) Probability of at most six people being unemployed:
To find this probability, we need to calculate the sum of probabilities for x = 0, 1, 2, 3, 4, 5, and 6.

P(X ≤ 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)

This calculation can be time-consuming if done manually. Using a calculator or software with statistical functions is recommended. The result is approximately 0.9671.

3) Probability of at least four people being unemployed:
To find this probability, we need to calculate the sum of probabilities for x = 4, 5, 6, ..., 100.

P(X ≥ 4) = P(X = 4) + P(X = 5) + P(X = 6) + ... + P(X = 100)

Again, using a calculator or software with statistical functions is recommended. The result is approximately 0.3291.

I hope this helps!