91Ó°ÊÓ

In probability, the complement of an event gives us the likelihood that the event does not happen. For any event F, the complement event (denoted as \(F^c\)) represents all outcomes that are not in F. The formula to calculate the probability of the complement event is quite straightforward. It states that the probability of the complement of an event is equal to 1 minus the probability of the event itself. Mathematically, this is represented as: \[ P(F^c) = 1 - P(F) \]
If you know the probability of an event occurring, you can quickly find out the probability of it not occurring by using this rule. It’s a handy tool because often it’s easier to calculate the probability of the complement. For example, imagine rolling a die. If the probability of rolling a six is \( \frac{1}{6} \), the probability of not rolling a six is \( 1 - \frac{1}{6} = \frac{5}{6} \). This demonstrates the complement rule in action.

Probability helps us measure how likely an event is to occur. It ranges from 0 to 1, where 0 means the event will not happen and 1 means it certainly will. Here’s how you can calculate probabilities:

• Identify all possible outcomes.
• Determine the number of favorable outcomes.
• Divide the number of favorable outcomes by the total number of possible outcomes.

For example, if there are 4 red balls and 6 blue balls in a bag, the probability of picking a red ball is \( \frac{4}{10} = 0.4 \).
When calculating the probability of the complement, like in the given exercise:
1. We start with the probability of the event F happening, which is given as \( P(F) = 0.45 \).
2. Then, we use the complement rule: \[ P(F^c) = 1 - P(F) \]
By substituting \( P(F) \) with 0.45, we get: \[ P(F^c) = 1 - 0.45 = 0.55 \]
So, the probability that event F does not happen (the complement) is 0.55. This means there is a 55% chance that event F will not occur.

An event in probability is any outcome or a set of outcomes from a random process. Events can be simple, like flipping a coin and getting heads, or complex, like drawing a red card from a shuffled deck of cards.
Events can be:

• Independent: Occurrence of one event does not affect the other (e.g., two successive coin tosses).
• Dependent: Occurrence of one event affects the other (e.g., drawing two cards in succession without replacement).

In the given exercise, we focus on the event F and its complement.
If event F represents something specific happening (like a rain forecast of 45% probability), \( F^c \) represents the opposite (not raining) with a probability of 55%, using our earlier calculation.
Understanding the nature of the event helps us apply probability rules accurately and effectively interpret our results.