Adding Probabilities 12-5
The Addition Rule – “Or”
1. The special addition rule (mutually
exclusive events)
2. The general addition rule (non-mutually
exclusive events)
Two computer simulations of tossing a balanced coin
100 times: The Law of Large Numbers
Relationships Among Events
(not E): The event that “E does not occur.”
(A & B): The event that “both A and B occur.”
(A or B): The event that “either A or B or both
occur.”
Combinations of Events
The Addition Rule – “Or”
• The special addition rule (mutually exclusive
events)
• The general addition rule (non-mutually exclusive
events)
Venn diagrams for
(a) event (not E)
(b) event (A & B)
(c) event (A or B)
Mutually Exclusive Events
Two or more events are said to be mutually exclusive if at
most one of them can occur when the experiment is performed,
that is, if no two of them have outcomes in common
Two mutually exclusive events
(a) Two mutually exclusive events
(b) Two non-mutually exclusive events (inclusive)
(a) Three mutually exclusive events (b) Three nonmutually exclusive events (c) Three non-mutually
exclusive events
Mutually Exclusive or Not?
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•
•
•
•
A flipped coin: heads or tails?
A student taking a test: passing the test and getting a B?
A student taking a test: passing the test and failing it?
When rolling a six-sided die, rolling a 3 or an odd number?
When rolling a six-sided die, rolling an even or an odd
number?
• Choosing a card from a standard deck and it is an ace or a
spade?
• Choosing a card from a standard deck and it is an ace or a
king?
The Special Addition Rule
If event A and event B are mutually exclusive, then
P A or B  P A  PB
More generally, if events A, B, C, … are mutually exclusive, then
P A or B or C ...  P A  PB  PC  ...
That is, for mutually exclusive events, the probability that at least one of
the events occurs is equal to the sum of the individual probabilities.
The General Addition Rule
If A and B are any two events, then
P(A or B) = P(A) + P(B) – P(A & B).
In words, for any two events, the
probability that one or the other occurs
equals the sum of the individual
probabilities less the probability that both
occur.
P(A or B): Spade or Face Card
P(A or B): Spade or Face Card
P (spade) + P (face card) – P (spade & face card) = 1/4 + 3/13 – 3/52
= 22/52
Summary
For the “Or” rule:
• if the events are not mutually exclusive you have to subtract off
the double count …
•
If the events are NOT mutually exclusive, the double count is
zero.
Either way, you can always use the formula:
P(A or B) = P(A) + P(B) – P(A & B).