Independent 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. The probability of one event does not change the probability of the other event.

If A and B are independent events, the probability of both events occurring is the product of the probabilities of the individual events.

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If A and B are independent events,

P(AB) = P(A and B) = P(A) • P(B).
(referred to as the "Probability Multiplication Rule")


Example 1:  A drawer contains 3 red paper clips, 4 green paper clips, and 5 blue paper clips.  One paper clip is taken from the drawer and then replaced.  Another paper clip is taken from the drawer.  What is the probability that the first paper clip is red and the second paper clip is blue?

Because the first paper clip is replaced, the sample space of 12 paper clips
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.



When you toss a coin, the probability of getting a head is 1 out of 2 or ½.
cointoss If you toss the coin again, the probability of getting a head is still 1 out of 2 or ½.
If you toss a coin 10 times and get a head each time, you may think that your luck of tossing a tail is increasing since it has not yet appeared. This is not the case.
These events are independent events and do not affect one another.
The probability of tossing a tail is 1 out of 2 or ½ regardless of how many heads were tossed previously.

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Dependent Events




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.

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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(AB) = P(A and B) = P(A) • P(B | A)

When dealing with dependent events, you are dealing with conditional probability.

Example:  A drawer contains 3 red paper clips, 4 green paper clips, and 5 blue paper clips.  One paper clip is taken from the drawer and is NOT replaced. Another paper clip is taken from the drawer.  What is the probability that the first paper clip is red and the second paper clip 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 paper clips, but the sample space of the second event is now 11 paper clips. 
The events are dependent.
P(red then blue) = P(red) • P(blue) = 3/12 • 5/11 = 15/132 = 5/44.


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reminder Be on the lookout for the word "replacement" as a clue.

With Replacement: the events are independent. Probabilities do not affect one another.
Without Replacement: the events are dependent. The second probability is affected.


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