Question

a local county has an unemployment rate of 4.6%. a random sample of 15 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 3 in the sample are unemployed.
c) find the probability that there are more than 4 in the sample are unemployed.
d) find the probability that there are at most 3 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 \(n = 15\), \(p=0.046\), \(1-p = 0.954\)

Step2: Solve part (a)

For \(k = 3\)

$$
LATEXBLOCK0
$$

Using a calculator, \(P(X = 3)\approx0.0180\)

Step3: Solve part (b)

\(P(X\lt3)=P(X = 0)+P(X = 1)+P(X = 2)\)
For \(k = 0\):

$$
LATEXBLOCK1
$$

For \(k = 1\):

$$
LATEXBLOCK2
$$

For \(k = 2\):

$$
LATEXBLOCK3
$$

\(P(X\lt3)=0.4990 + 0.3677+0.1133 = 0.98 00\)

Step4: Solve part ( c)

\(P(X\gt4)=1-(P(X = 0)+P(X = 1)+P(X = 2)+P(X = 3)+P(X = 4))\)
For \(k = 4\):

$$
LATEXBLOCK4
$$

Using a calculator, \(P(X = 4)\approx0.0010\)
\(P(X\gt4)=1-(0.4990 + 0.3677+0.1133+0.0180+0.0010)=0.0010\)

Step5: Solve part (d)

\(P(X\leq3)=P(X = 0)+P(X = 1)+P(X = 2)+P(X = 3)\)
\(P(X\leq3)=0.4990+0.3677 + 0.1133+0.0180=0.9980\)

Answer:

a) \(0.0180\)
b) \(0.9800\)
c) \(0.0010\)
d) \(0.9980\)