Problem 8 - Algebra 2

Question

\(\boldsymbol{Q}\) and \(\boldsymbol{R}\) are independent events. Find \(\boldsymbol{P}(\boldsymbol{Q} \text { and } \boldsymbol{R})\) $$ P(Q)=\frac{1}{3}, P(R)=\frac{6}{x} $$

Step-by-Step Solution

Answer

\(P(Q \text { and } R) =\frac{6}{3x}\) which simplifies to \( \frac{2}{x}\)

Step 1: Identify Probabilities of Individual Events

In the problem, the probability of event Q, denoted \( P(Q) \), is stated to be 1/3. The probability of event R, denoted \( P(R) \), is given as 6/x, where x is a constant number.

Step 2: Apply the Formula for Independent Events

Since events Q and R are independent, the probability of both events occurring together, symbolized as \( P(Q \text { and } R) \), can be found by multiplying the probabilities of each individual event. Therefore, we can write the formula as follows: \( P(Q \text { and } R) = P(Q) \times P(R) \)

Step 3: Calculate the Probability of Q and R

Using the formula from Step 2, plug the values of \( P(Q) \) and \( P(R) \) into it to find \( P(Q \text { and } R) \). Hence, \( P(Q \text { and } R) = \frac{1}{3} \times \frac{6}{x} = \frac{6}{3x} \).

Key Concepts

Independent Events

In the world of probability, events are considered independent if the occurrence or non-occurrence of one event does not affect the probability of the other event occurring. This means that the outcome of one event has no influence on the outcome of another event. For example, flipping a coin and rolling a die are independent events since the result of each does not depend on the other. Understanding this concept is crucial because it helps us determine how to calculate the combined probabilities of multiple events.

Independent events are foundational in probability theory and your calculations of such events involve a straightforward process thanks to their lack of dependence on one another. By recognizing two events as independent, you simplify the process of calculating their joint probability, making it just a matter of using a specific formula.

Probability Formula

Probability formulas are the mathematical tools used for determining the chance of a particular outcome. When dealing with probability, you typically are looking to find the likelihood that an event will occur, which is represented in a numerical form ranging from 0 to 1. A probability of 0 means the event will never occur, whereas a probability of 1 indicates the event will always occur. Most real-world probabilities fall between these extremes.

Formally, probability is defined as the number of successful outcomes divided by the total number of possible outcomes. For example, the probability of flipping a heads with a standard coin is calculated as 1 (one head on the coin) divided by 2 (the total number of sides), resulting in a probability of 0.5 or 50%. By mastering these formulas, you can predict and prepare for various scenarios in everything from games to daily decision-making.

Multiplication Rule for Probabilities

When it comes to calculating the probability of two independent events happening together, you use the multiplication rule for probabilities. This rule states that if two events, say A and B, are independent, the probability of both events occurring is the product of their individual probabilities.

To apply the multiplication rule, you simply multiply the probability of the first event by the probability of the second event, as shown in this formula:

• \[ P(A \text{ and } B) = P(A) \times P(B) \]

This straightforward approach relies entirely on the independence of the events, meaning one event's occurrence does not influence the other. For instance, if you know the probability of flipping a coin and getting heads is 0.5 and the probability of rolling a die and getting a 4 is 1/6, the probability of both events happening together (getting heads and rolling a 4) is \( 0.5 \times \frac{1}{6} = \frac{1}{12} \), or approximately 0.0833.

Understanding and applying the multiplication rule makes it simpler to calculate joint probabilities for more complex scenarios, especially when analyzing real-world cases where outcomes are interdependent.