Problem 7 - Algebra 2

Question

\(\boldsymbol{Q}\) and \(\boldsymbol{R}\) are independent events. Find \(\boldsymbol{P}(\boldsymbol{Q} \text { and } \boldsymbol{R})\) $$ P(Q)=0.6, P(R)=0.9 $$

Step-by-Step Solution

Answer

The probability of both Q and R occurring, given that they are independent events, is 0.54.

Step 1: Understand the Definition of Independent Events

Understand that two events Q and R are independent if the occurrence of one does not affect the probability of the occurrence of the other. The probability of both independent events happening is given by the product of their individual probabilities, i.e., P(Q and R) = P(Q) * P(R).

Step 2: Identify the Given Probabilities

From the given exercise, identify the probabilities of the independent events Q and R. Here, P(Q) = 0.6 and P(R) = 0.9.

Step 3: Compute P(Q and R) for Independent Events

On understanding the concept of independent events and identifying the given probabilities, the next step is to find the probability of both events occurring. This can be done by multiplying the probabilities of events Q and R. So, P(Q and R) = P(Q) * P(R) = 0.6 * 0.9.

Key Concepts

Probability

In everyday language, probability measures how likely an event is to occur. For instance, the likelihood of it raining might be expressed as a probability. In mathematical terms, probability is a number between 0 and 1.

• 0 means the event will not happen.
• 1 means the event is certain to happen.

A coin toss is a classic example with a probability of 0.5 for getting either heads or tails. It’s essential to grasp this concept for events like

• Q
• R

from the exercise at hand, to then comfortably multiply and find desired outcomes.

Multiplication Rule

The multiplication rule is crucial when dealing with probabilities of independent events. When two events are independent, the probability of both occurring together is the product of their probabilities. This is why we can use the multiplication rule conveniently here.For example, if you know:

• Event Q has a probability of 0.6
• Event R has a probability of 0.9

Then the combined probability of both events occurring at the same time—P(Q and R)—can be found via:\[P(Q \text{ and } R) = P(Q) \times P(R)\]In this exercise, applying this rule means calculating 0.6 multiplied by 0.9, resulting in 0.54. This shows that there is a 54% chance both events will occur together.

Conditional Probability

Conditional probability ponders the likelihood of an event occurring when another related event has already happened. However, in the context of independent events like

• Q
• R

in our example, it's irrelevant as the occurrence of one event offers no insight into the occurrence of the other. Though understanding conditional probability is less critical in this scenario, it's important to know it contrasts starkly with independent events where outcomes do affect each other. In problems where events are dependent, you'd use conditional probability to adjust expectations and calculations.

Event Occurrence

An event occurrence refers to the actual happening of a given event. In probability theory,

• Each potential event, whether it occurs or not, has a probability assigned to it.
• Events are considered independent if the occurrence of one does not change the probability of the other happening.

In the case of events Q and R, being independent implies that calculating their combined occurrence is straightforward with the use of multiplication. Understanding this effect simplifies a lot of complex probability challenges because it allows you to calculate large-scale event compound probabilities by merely multiplying individual probabilities. It reminds us that probability theory powers through simple arithmetic principles when dealing with independent events.