Problem 63 - Algebra and Trigonometry

Question

Explaining the Concepts Explain how to find or probabilities with events that are not mutually exclusive. Give an example.

Step-by-Step Solution

Answer

To find the or probability of non-mutually exclusive events, you add the individual probabilities of each event and subtract the probability of the events occurring together. Example: for a die roll, the likelihood of getting either an even number or a number less than five is calculated by \(P(A \cup B) = P(A) + P(B) - P(A \cap B) = 1/2 + 2/3 - 1/3 = 5/6\). This means in 5 out of 6 cases, an outcome satisfying either or both conditions will be achieved.

Step 1: Understand Non-Mutually Exclusive Events

Non-mutually exclusive events can occur at the same time. In other words, the occurrence of one event does not preclude the occurrence of another. A typical example is tossing a die: getting an even number (2, 4, 6) or getting a number less than 5 (1, 2, 3, 4) are not mutually exclusive - there are outcomes that satisfy both.

Step 2: Learn the Formula for Or Probabilities for Non-Mutually Exclusive Events

The formula is \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\), where \(P(A \cup B)\) describes the probability of either event A or event B occurring, \(P(A)\) and \(P(B)\) respectively describe the probabilities of events A and B, and \(P(A \cap B)\) describes the probability of both event A and event B occurring together. The subtraction of \(P(A \cap B)\) is to avoid counting the possibilities where A and B occur together twice.

Step 3: Example of Calculating the Or Probability for Non-Mutually Exclusive Events

Consider the roll of a die. Define the two events as follows: Event A: The die shows an even number (2, 4, or 6). Event B: The die shows less than 5 (1, 2, 3 or 4). The probabilities for each event are: \(P(A) = 1/2\), \(P(B) = 2/3\), \(P(A \cap B) = 1/3\). So from our second step, the probability for A or B is \(P(A \cup B) = P(A) + P(B) - P(A \cap B) = 1/2 + 2/3 - 1/3 = 5/6\). There is a likelihood of 5 out of 6 that the roll of the die will result in a number that is either even or less than 5 (or both).

Key Concepts

Probability

Probability is a fascinating branch of mathematics. It helps us predict how likely events are to occur. The probability of an event is a number between 0 and 1, where 0 indicates impossibility and 1 signifies certainty.

To calculate the probability of an event, we use the formula:

• Probability of event = (Number of favorable outcomes) / (Total number of possible outcomes).

For example, if you toss a fair six-sided die, the chance of it landing on a specific number, like 3, will be 1 out of 6. This is because there is one favorable outcome (rolling a 3) and a total of 6 possible outcomes (1, 2, 3, 4, 5, 6).
Understanding basic probability sets the stage for more complex concepts like non-mutually exclusive events and their calculations.

Or probability

Or probability, also known as the union of probabilities, tells us how likely it is for at least one of multiple events to occur. It's especially useful when dealing with events that are not mutually exclusive.

For example, if we're rolling a die and we want to know the probability of rolling a number that's either even or less than 5, we need to consider both possibilities. In this case, the formula changes slightly to ensure we don't count any outcomes twice.

The concept of "or" allows us to expand our investigation, considering multiple scenarios in a single probability calculation. This is different from "and" probability, which seeks the probability of multiple events occurring simultaneously.

Probability formula

For non-mutually exclusive events, calculating the probability requires a specific formula:

• \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Here, \( P(A \cup B)\) is the probability of either event A or event B occurring.

\( P(A)\) and \(P(B)\) are the probabilities of each individual event.
\( P(A \cap B)\) is the probability of both events happening at the same time, and we subtract it to avoid double counting. This formula is crucial for non-mutually exclusive events. It ensures accuracy in scenarios where outcomes can satisfy multiple conditions at once. Knowing this formula can greatly help in solving many probability exercises with accuracy and ease.

Math example

Let's solidify this understanding with a math example using a common six-sided die. Suppose two events occur:

• Event A: Rolling an even number (2, 4, 6),
• Event B: Rolling a number less than 5 (1, 2, 3, 4).

To find \( P(A)\): There are 3 even numbers (2, 4, 6), so \( P(A) = \frac{3}{6} = \frac{1}{2} \).
For \( P(B)\): There are 4 numbers less than 5 (1, 2, 3, 4), so \( P(B) = \frac{4}{6} = \frac{2}{3} \).
Both being less than 5 and even (2, 4) happen twice, so \( P(A \cap B) = \frac{2}{6} = \frac{1}{3} \).
Using the formula for or probability: \[ P(A \cup B) = \frac{1}{2} + \frac{2}{3} - \frac{1}{3} = \frac{5}{6} \].
This result means there is a high chance, specifically a 5 out of 6 chance, that a number that satisfies either condition will be rolled.