Question

the following data were gathered for 110 workers in two departments. of these workers, 49 were in department a, 6 workers in department a earned more than $80,000, and 34 workers in department b earned $80,000 or less. complete parts (a) through (d) below.
(a) find the probability that a worker is in department a earning $80,000 or less.
0 39
(type an integer or decimal rounded to two decimal places as needed.)
(b) find the probability that a worker is in department b earning more than $80,000
0.25
(type an integer or decimal rounded to two decimal places as needed.)
(c) find the probability that a worker is in department b or is earning more than $80,000

(type an integer or decimal rounded to two decimal places as needed.)

Explanation:

Step1: Calculate the number of workers in Department B

The total number of workers is 110 and 49 are in Department A. So the number of workers in Department B is $110 - 49=61$.

Step2: Calculate the number of workers in Department B earning more than $80,000$

We know that 34 workers in Department B earned $80,000$ or less. So the number of workers in Department B earning more than $80,000$ is $61 - 34 = 27$.

Step3: Calculate the number of workers earning more than $80,000$ in total

6 workers in Department A earned more than $80,000$ and 27 workers in Department B earned more than $80,000$, so the total number of workers earning more than $80,000$ is $6 + 27=33$.

Step4: Use the formula for $P(A\cup B)$

The formula for $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Let $A$ be the event that a worker is in Department B and $B$ be the event that a worker is earning more than $80,000$.
The number of workers in Department B is 61, the number of workers earning more than $80,000$ is 33, and the number of workers in Department B earning more than $80,000$ is 27.
$P(A)=\frac{61}{110}$, $P(B)=\frac{33}{110}$, $P(A\cap B)=\frac{27}{110}$.
$P(A\cup B)=\frac{61 + 33- 27}{110}=\frac{67}{110}\approx0.61$

Answer:

$0.61$