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Complementary Event

A complementary event refers to the exact opposite of a specific event occurring within a random experiment. If an event represents a certain outcome, its complementary event includes all other possible outcomes in the sample space where that specific event does not happen.

Key Concepts

Every event E has a corresponding complementary event, usually denoted as 'not E', E', or E with a bar over it.
The sum of the probabilities of an event and its complementary event is always exactly 1.
Complementary events are mutually exclusive and exhaustive, meaning they cannot occur at the same time and together they cover every possible outcome of the experiment.

Formula & Equation

P(E)+P(E‾)=1 or P(E‾)=1−P(E)P(E) + P(\overline{E}) = 1 \text{ or } P(\overline{E}) = 1 - P(E)

P(E) is the probability of event E occurring, and P(\overline{E}) is the probability of the complementary event 'not E' occurring.

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Common Misconceptions

Myth: Complementary events always have an equal 50-50 chance of occurring.
Fact: Complementary events only have equal probabilities if the original event has a 0.5 probability. For example, if the chance of winning a game is 10 percent, the complementary event of not winning is 90 percent.
Myth: Any two mutually exclusive events are complementary.
Fact: While mutually exclusive events cannot happen at the same time, they are only complementary if their probabilities add up to exactly 1, meaning together they cover the entire sample space.

Real World Applications

When rolling a standard six-sided die, if the event is rolling a 4, the complementary event is rolling a 1, 2, 3, 5, or 6.
If the weather forecast states the probability of rain tomorrow is 0.3, the complementary event is it not raining tomorrow, which has a probability of 0.7.
When drawing a single card from a standard deck, if the event is drawing a face card, the complementary event is drawing any card that is not a face card.

Frequently Asked Questions

What is the sum of the probabilities of an event and its complementary event?
The sum of the probability of an event and its complementary event is always equal to 1.
How do you calculate the probability of a complementary event?
You calculate the probability of a complementary event by subtracting the probability of the original event from 1.
Can a complementary event have a probability greater than 1?
No, because the probability of any event, including a complementary event, must always lie between 0 and 1 inclusive.

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