Question

During a 52 week period, a company paid overtime wages for 19 weeks and hired temp help for 11 weeks. During 5 weeks, the company paid overtime AND hired temp help. If an auditor randomly examined the payroll for only one week, whats the probability that the payroll for that week contained overtime wages or temp help wages? ...?

Answer 1

The probability that the payroll for one week contained overtime wages or temporary help is $\boxed{\frac{10}{13}}$.

Further explanation:

It is given that during a $52$ week period, a company paid overtime wages for $19$ weeks and hired temporary help for $11$ weeks.

During $5$ weeks, the company paid overtime and hired temporary help.

Calculation:

A company paid overtime wages for $19$ weeks and $5$ weeks during the $52$ week period.

Therefore, the probability of overtime wages $P(O)$ is calculated as follows:

$\begin{aligned}P(O)&=\dfrac{19}{52}+\dfrac{5}{52}\\&=\dfrac{19+5}{52}\\&=\dfrac{24}{52}\end{aligned}$

The probability of overtime wages is $\frac{24}{52}$.  

The same company hired temporary help for $11$ weeks and $5$ weeks during the $52$ week period.

Therefore, the probability of temporary help $P(H)$ is calculated as follows:

$\begin{aligned}P(H)&=\dfrac{11}{52}+\dfrac{5}{52}\\&=\dfrac{11+5}{52}\\&=\dfrac{16}{52}\end{aligned}$  

The probability of temporary help is $\frac{16}{52}$.  

Now, the combined probability $P$ that the payroll for one week contained overtime wages or temporary help is calculated by adding the probabilities of overtime wages and temporary help as follows:

$\begin{aligned}P&=\dfrac{24}{52}+\dfrac{16}{52}\\&=\dfrac{24+16}{52}\\&=\dfrac{40}{52}\\&=\dfrac{10}{13}\end{aligned}$

 

The probability that the payroll for one week contained overtime wages or temporary help is $\boxed{\dfrac{10}{13}}$.

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Answer details:

Grade: Middle school

Subject: Mathematics

Chapter: Probability

Keywords: Probability, events, outcome, sample, experiment, equally likely, random, complementary events, joint probability, independent events, mutually exclusive events, conditional probability.

Answer 2

P(overtime wage) = 19/52 + 5/52 = 24/52
P(temp help wage) = 11/52 + 5/52 = 16/52
P(overtime or temp help) = P(overtime) + P(temp help)
= 24/52 + 16/52
= 40/52 = 10/13