6.3: Complementary Events .
- Page ID
- 139282
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Now let us examine the probability that an event does not happen. As in the previous section, consider the situation of rolling a six-sided die and first compute the probability of rolling a six: the answer is \(P(\operatorname{six})=1 / 6\). Now consider the probability that we do not roll a six: there are 5 outcomes that are not a six, so the answer is \(P(\) not a six \()=\dfrac{5}{6}\).
Notice that \(P(\operatorname{six})+P(\) not a six \()=\dfrac{1}{6}+\dfrac{5}{6}=\dfrac{6}{6}=1 \quad\) This is not a coincidence.
The complement of an event is the event " \(E\) does NOT happen".
The notation \(\bar{E}\) is used for the complement of event \(E\). You should recognize this notation from the last chapter.
For any event \(E, P(E)+P(\bar{E})=1\)
We can compute the probability of the complement using \(P(\bar{E})=1-P(E)\)
Notice also that \(P(E)=1-P(\bar{E})\)
If you pull a random card from a deck of playing cards, what is the probability it is not a heart?
Solution
There are 13 hearts in the deck, so \(P\) (heart) \(=\dfrac{13}{52}=\dfrac{1}{4}\).
The probability of not drawing a heart is the complement:
\[
P(\text { not heart })=1-P(\text { heart })=1-\dfrac{1}{4}=\dfrac{3}{4}
\]
A jar contains 28 marbles, 12 of which are red. If you pick 1 marble out of the jar, what is the probability that it is not red?
Solution
There are 12 red marbles out of 28 total marbles, so \(\mathrm{P}(\mathrm{red})=\dfrac{12}{28}=\dfrac{3}{7}\).
The probability of not drawing a red marble is the complement:
\[
P(\text { not red })=1-\dfrac{3}{7}=\dfrac{4}{7}
\]
Your favorite basketball player is an 84% free throw shooter. Find the probability that they do NOT make their next free throw.
- Answer
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1-0.84 = 0.16