Chapter 3. Probability: Probability
Probability of the Complement
Besides calculating the probability that event #A# will happen, you can also calculate the probability that #A# will not happen. In other words: the probability that the complement of A happens.
Rules
- The probability that an event #A# happens plus the probability that its complement #A^c# happens is always equal to #1#.
#\mathbb{P}(A) + \mathbb{P}(A^c) = 1# - Complement Rule:
#\mathbb{P}(A^c) = 1 - \mathbb{P}(A)# - Subtraction Rule:
#\mathbb{P}(A) = 1 - \mathbb{P}(A^c)#
Sometimes when calculating the probability of event #A#, it is easier to calculate the probability of its complement #A^c# first.
Observing two different numbers would for example be rolling a #2# and a #3#, or a #1# and a #6#. This makes for quite a long list of possible outcomes:
\[A= \{(1,2)(1,3),(1,4),(1,5),(1,6),(2,1),\ldots, (6,5)\}\]
But the complement of this event, #A^c# (the event that two numbers are the same), only contains #6# outcomes:
\[A^c = \{(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)\}\]
The sample space of this random experiment contains a total of #6^2=36# possible outcomes. The probability of #A^c# occurring is thus calculated as:
\[\mathbb{P}(A^c)=\cfrac{\text{number of outcomes where the two numbers are the same}}{\text{total number of possible outcomes}}=\cfrac{6}{36}=\cfrac{1}{6}\]
Now, by applying the complement rule, the probability of #A# can be calculated:
\[\mathbb{P}(A)=1-\mathbb{P}(A^c)=1-\cfrac{1}{6}=\cfrac{5}{6}\]
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