Question

a local county has an unemployment rate of 4.1%. a random sample of 16 employable people are picked at random from the county and are asked if they are employed. round answers to 4 decimal places.
a) find the probability that exactly 3 in the sample are unemployed.
b) find the probability that there are fewer than 4 in the sample are unemployed.
c) find the probability that there are more than 2 in the sample are unemployed.
d) find the probability that there are at most 4 in the sample are unemployed.
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Explanation:

Step1: Identify the binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n\) is the number of trials, \(k\) is the number of successes, \(p\) is the probability of success in a single trial, and \(C(n,k)=\frac{n!}{k!(n - k)!}\)

Given \(n = 16\), \(p=0.041\), \(1-p = 0.959\)

Step2: Calculate part (a)

For \(k = 3\)

$$
LATEXBLOCK0
$$

Step3: Calculate part (b)

\(P(X\lt4)=P(X = 0)+P(X = 1)+P(X = 2)+P(X = 3)\)

For \(k = 0\):

$$
LATEXBLOCK1
$$

For \(k = 1\):

$$
LATEXBLOCK2
$$

For \(k = 2\):

$$
LATEXBLOCK3
$$

\(P(X = 3)\approx0.0229\) (from part (a))

\(P(X\lt4)=0.5137 + 0.3521+0.1253 + 0.0229=0.9140\)

Step4: Calculate part (c)

\(P(X\gt2)=1 - P(X\leq2)=1-(P(X = 0)+P(X = 1)+P(X = 2))\)

\(P(X = 0)\approx0.5137\), \(P(X = 1)\approx0.3521\), \(P(X = 2)\approx0.1253\)

\(P(X\gt2)=1-(0.5137 + 0.3521+0.1253)=1 - 0.9911=0.0089\)

Step5: Calculate part (d)

\(P(X\leq4)=P(X = 0)+P(X = 1)+P(X = 2)+P(X = 3)+P(X = 4)\)

For \(k = 4\):

$$
LATEXBLOCK4
$$

\(P(X\leq4)=0.9140+0.0032 = 0.9172\)

Answer:

a) \(0.0229\)
b) \(0.9140\)
c) \(0.0089\)
d) \(0.9172\)