Understanding probability involves not only calculating the likelihood of an event occurring but also determining the probability that an event will not happen. This concept is known as the complement of an event. For instance, when rolling a six-sided die, if we define the event of rolling a four as event A, the complement of A includes all outcomes where a four is not rolled, which are rolling a one, two, three, five, or six.
The notation for the complement of an event can vary; it may be represented as \( A' \), \( \overline{A} \), or \( \neg A \). To calculate the probability of event A occurring, we use the formula:
\[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}} \]
In the case of rolling a four, there is one favorable outcome (rolling a four) out of six possible outcomes, so:
\[ P(A) = \frac{1}{6} \]
To find the probability of not rolling a four, we can count the five outcomes that represent the complement of A. Thus, the probability of the complement of A is:
\[ P(A') = \frac{5}{6} \]
It is important to note that the sum of the probabilities of an event and its complement always equals one:
\[ P(A) + P(A') = 1 \]
This leads us to a useful formula for calculating the probability of an event not happening:
\[ P(A') = 1 - P(A) \]
For example, when drawing a card from a standard deck of 52 cards, if we want to find the probability of not drawing a queen, we first calculate the probability of drawing a queen. There are four queens in the deck, so:
\[ P(\text{Queen}) = \frac{4}{52} \]
Using the complement formula, we find the probability of not drawing a queen:
\[ P(\text{Not Queen}) = 1 - P(\text{Queen}) = 1 - \frac{4}{52} = \frac{48}{52} \approx 0.92 \]
This means there is a 92% chance of not drawing a queen from a standard deck of cards. By applying the concept of complements, we can simplify our calculations and gain a clearer understanding of probability.
