Two events are independent if the occurrence of one event has no impact on the occurrence of the other event. The definition for two independent events is:
\begin{equation*}
P(A|B)=P(A)
\end{equation*}
Events are dependent if the occurrence of one event affects the occurrence of another event.
Figure4.2.2.(Made in GeoGebra by Steve Phelps)
Exercise4.2.3.
A delivery service operates in a city where it snows in the winter. Suppose it is now winter. We consider the events that on a randomly selected day that they are late making deliveries, and the event that it snows that day.
Suppose that the probability that the service is late on this day is \(0.17\text{,}\) the probability that the service is late given that it snows on this day is \(0.7\text{,}\) and finally the probability that it snows on this day is \(0.18\text{.}\)
(a)
Find the probability that the service is late and it snows on this day.
Answer.
The probability that the service is late and it snows on this day is \(0.126\text{.}\)
(b)
Find the probability that it snows, given that the service is late on this day.
Answer.
The probability that it snows, given that the service is late on this day is \(0.7411764705882352\text{.}\)
(c)
Are the events the service is late on this day, and it snows on this day independent? Explain your reasoning.
Answer.
The events the service is late on this day, and it snows on this day are not independent because the probability that it snows, given that service is late is not equal to the probability that it snows.
(The events are dependent.)
The multiplication rule for probabilities is used to calculate the probability for intersection (or joint probability) of two events occurring.
It depends on knowing whether or not two events are independent or dependent.
A company has 140 employees, of which 30 are supervisors. Eighty of the employees are married, and 20% of the married employees are supervisors. If a company employee is randomly selected, what is the probability that employee is married and is a supervisor?