How to Find Probability of Simple and Opposite Events
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In probability theory, understanding simple and opposite events is essential. This guide will walk you through determining their probabilities step by step.
Simple Events
A simple event is an event that describes a single outcome.
Step 1: Understand the Scenario
Before diving into the calculations, read the problem carefully and understand the scenario.
Step 2: Determine the Total Number of Possible Outcomes
This is often represented by the denominator when expressing probability. For instance, when flipping a coin, there are 2 possible outcomes: heads or tails.
Step 3: Identify the Desired Outcome
This will be the numerator. Using the coin flip example, if you're trying to find the probability of getting heads, the desired outcome is 1.
Step 4: Calculate the Probability
The probability \( P \) of an event \( E \) is calculated as:
\[ P(E) = \frac{\text{number of desired outcomes}}{\text{total number of possible outcomes}} \]
Opposite Events
The opposite (or complementary) event of an event \( E \) is the event "not \( E \)".
Step 1: Find the Probability of the Given Event
Using the formula from above, find the probability of event \( E \).
Step 2: Determine the Opposite Event's Probability
The probability of the opposite event is given by:
\[ P(\text{not } E) = 1 - P(E) \]
For example, if the probability of drawing a red card from a standard deck is \( \frac{26}{52} \), then the probability of not drawing a red card is \( 1 - \frac{26}{52} = \frac{26}{52} \).
Conclusion
Understanding the probability of simple and opposite events allows you to better predict outcomes based on given information. With practice, these calculations will become second nature.
Example
Simple Events
A simple event refers to a single outcome of a random experiment. The probability of a simple event is given by:
\[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]
Example:
What is the probability of rolling a 5 on a fair six-sided die?
Since there's only 1 way to roll a 5 and 6 possible outcomes:
\[ P(5) = \frac{1}{6} \]
Opposite Events
The opposite (or complementary) event of \( A \) is the event that \( A \) does not occur. It's denoted as \( A' \) or \( \bar{A} \).
The probability of an opposite event is given by:
\[ P(A') = 1 - P(A) \]
Example:
Using the die example, what is the probability of not rolling a 5 on a fair six-sided die?
Using the probability we found earlier:
\[ P(\text{not } 5) = 1 - \frac{1}{6} = \frac{5}{6} \]
Now, you're equipped to tackle problems related to simple and opposite event probabilities. Practice with more examples to strengthen your understanding.
