How to Find Probability of Simple and Opposite Events

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.

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.