Problem 6 - Algebra 2

Question

\(\boldsymbol{Q}\) and \(\boldsymbol{R}\) are independent events. Find \(\boldsymbol{P}(\boldsymbol{Q} \text { and } \boldsymbol{R})\) $$ P(Q)=\frac{12}{17}, P(R)=\frac{3}{8} $$

Step-by-Step Solution

Answer

The probability of both events Q and R occurring is \(\frac{9}{34}\)

Step 1: Identify the given probabilities

In this problem, we are given the probabilities of events Q and R. Specifically, we are told that \(P(Q) = \frac{12}{17}\) and \(P(R) = \frac{3}{8}\)

Step 2: Apply the rule for independent events

Since Q and R are independent events, their joint probability is the product of their individual probabilities. We can calculate this using the formula \(P(Q \text{ and } R) = P(Q) \times P(R)\)

Step 3: Calculate the joint probability

Substitute the given probabilities into the formula: \(P(Q \text{ and } R) = \frac{12}{17} \times \frac{3}{8}\)

Step 4: Solve the multiplication

Finally, multiply the fractions to get the joint probability: \(P(Q \text{ and } R) = \frac{36}{136} = \frac{9}{34}\)

Key Concepts

Joint Probability

In the world of probability, there's a fascinating concept known as "joint probability." This refers to the probability of two events happening at the same time. When we talk about joint probability, we want to know how likely it is for both events, say Event Q and Event R, to occur together.
For example, if you are drawing two cards, one after the other, from a deck and you want to know the chance that both are hearts, you're asking for the joint probability of two "draw heart" events. In mathematical terms, joint probability is denoted as \(P(Q \text{ and } R)\) and is calculated differently based on whether the events are independent or dependent. Understanding the joint probability helps in making insights into scenarios involving multiple events simultaneously, which is a common need in many real-world situations. When you know how to find this combined probability, you gain insights into the full picture of complicated processes.

Independent Events

Independent events are a vital concept to grasp in probability. Two events are defined as independent if the occurrence of one event does not influence the occurrence of the other.
This characteristic simplifies calculating their joint probability, which is critical in solving more complex probability problems.
A good example can be flipping a coin and rolling a die. Whether the coin lands on heads or tails does not affect what number the die shows. Therefore, each result is independent of the other. This lack of influence is key to defining independent events. In practice, recognizing independent events allows us to calculate joint probabilities by straightforward multiplication. Understanding the nature of event independence gives you a sharper toolset for analyzing real-world situations and problems that involve multiple outcomes or activities happening at once.

Multiplication of Probabilities

When dealing with independent events, the way to calculate their joint probability is through the multiplication of probabilities. This is due to the independence property, which dictates that the occurrence of one event does not impact the likelihood of the other. To determine the probability of both events Q and R happening together, represented as \(P(Q \text{ and } R)\), simply multiply their individual probabilities: \(P(Q) \times P(R)\).
This is a straightforward process because, for independent events, their probabilities do not interfere with each other. In our specific problem, \(P(Q) = \frac{12}{17}\) and \(P(R) = \frac{3}{8}\). Multiplying these values gives the joint probability: \[P(Q \text{ and } R) = \frac{12}{17} \times \frac{3}{8} = \frac{36}{136} = \frac{9}{34}\] This formula is a powerful tool, as it lets you determine the chance of multiple independent events happening at the same time with simple arithmetic, helping simplify seemingly complex problems.