Independent and Dependent Events
Two events are said to be independent if the result of the second event is not affected by the result of the first event.
If A and B are independent events, the probability of both events occurring is the product of the probabilities of the individual events
If A and B are independent events,
P(A and B) = P(A) • P(B).
Example: A drawer contains 3 red paperclips, 4 green paperclips, and 5 blue paperclips. One paperclip is taken from the drawer and then replaced. Another paperclip is taken from the drawer. What is the probability that the first paperclip is red and the second paperclip is blue?
Because the first paper clip is replaced, the sample space of 12 paperclips does not change from the first event to the second event. The events are independent.
P(red then blue) = P(red) • P(blue) = 3/12 • 5/12 = 15/144 = 5/48.
If the result of one event is affected by the result of another event, the events are said to be dependent.
If A and B are dependent events, the probability of both events occurring is the product of the probability of the first event and the probability of the second event once the first event has occurred.
If A and B are dependent events, and A occurs first,
P(A and B) = P(A) • P(B,once A has occurred)
… and is written as …
P(A and B) = P(A) • P(B|A)
Example: A drawer contains 3 red paperclips, 4 green paperclips, and 5 blue paperclips. One paperclip is taken from the drawer and is NOT replaced. Another paperclip is taken from the drawer. What is the probability that the first paperclip is red and the second paperclip is blue?
Because the first paper clip is NOT replaced, the sample space of the second event is changed. The sample space of the first event is 12 paperclips, but the sample space of the second event is now 11 paperclips. The events are dependent.
P(red then blue) = P(red) • P(blue) = 3/12 • 5/11 = 15/132 = 5/44.