If A and B are overlapping events, they have outcomes in common. Therefore, P(AorB) = P(A)+P(B)-P(AandB).
1/50 or 0.02
Practice makes perfect
Let event A be selecting an employee who works part time and event B be selecting an employee who works 5 days each week. We are interested in finding the probability that a randomly selected employee works both part time and 5 days each week.
P(A and B)
We know there are 14 employees who work part time. Therefore, we can find P(A) using the theoretical probability. We need to compare the number of favorable outcomes to the number of possible outcomes, which is 100 — the total number of employees at the company.
P(A) = favorable outcomes/possible outcomes = 14/100Moreover, we know there are 80 employees who work 5 days each week. We can use this to obtain P(B).
P(B) = favorable outcomes/possible outcomes = 80/100
We are also given that 92 employees either work part time or 5 days each week.
P(A orB) = 92/100
Since we are interested in finding the probability that a randomly selected employee works both part time and 5 days each week, the events A and B are overlapping. Therefore, we can use the following formula.
P(AorB) = P(A) + P(B) - P(AandB)
We can substitute the obtained probabilities into this equation and solve for P(AandB).
Finally, we found the probability that a randomly selected employee works both part time and 5 days each week is 150 or 0.02.
Extra
Overlapping Events
To see the relation between two overlapping events, A and B, consider the following Venn diagram.
When two events are overlapping, P(A) contains a section of P(B), and vice versa. We call P(AandB) the overlap of this shared section. When we add the probabilities of the two events, we are adding this section twice. To fix this issue, we subtract P(AandB) once.
P(AorB) = P(A) + P(B) - P(AandB)
