Problem 5 - Algebra 2

Question

\(\boldsymbol{Q}\) and \(\boldsymbol{R}\) are independent events. Find \(\boldsymbol{P}(\boldsymbol{Q} \text { and } \boldsymbol{R})\) $$ P(Q)=\frac{1}{4}, P(R)=\frac{2}{3} $$

Step-by-Step Solution

Answer

\The probability of Q and R occurring together is \(P(Q \text{ and } R) = \frac{1}{6}\).

Step 1: Identify the Given Properties

From the problem, we know that the events Q and R are independent. Probability of event Q, \(P(Q) = \frac{1}{4}\) and the probability of event R, \(P(R) = \frac{2}{3}\).

Step 2: Apply the Formula for Probability of Independent Events

The formula for the probability of independent events Q and R occurring together is \(P(Q \text{ and } R) = P(Q) \cdot P(R)\).

Step 3: Calculate the Probability of Q and R

Having \(P(Q) = \frac{1}{4}\) and \(P(R) = \frac{2}{3}\), we need to multiply these probabilities. Hence, \(P(Q \text{ and } R) = \frac{1}{4} \cdot \frac{2}{3}\).

Key Concepts

Understanding Probability

Probability is a central concept in statistics that quantifies the likelihood of an event occurring. Imagine it as the measure of chance something will happen. It's usually expressed as a number between 0 and 1.

A probability of 0 means an event will not occur, while a probability of 1 means it will happen for sure. For example, if you flip a fair coin, the probability of it landing on heads is 0.5.

• If we say the probability of event Q is 1/4, it means that out of 4 trials, Q is expected to occur once on average.
• If event R has a probability of 2/3, it will likely happen in two-thirds of all trials.

In probabilistic terms, smaller numbers indicate that an event is less likely to happen, while numbers close to 1 suggest a higher likelihood of occurrence.

Applying the Multiplication Rule for Independent Events

The multiplication rule is key when dealing with independent events. Independent events are events where the result of one event does not affect the other. For instance, tossing a coin and rolling a die are independent events because the outcome of the coin toss doesn't change the result of the die roll.

The multiplication rule states that if two events are independent, the probability of both events occurring together is the product of their individual probabilities. In equation form, if Q and R are independent events, then:\[ P(Q \text{ and } R) = P(Q) \times P(R) \]

This formula helps in solving problems where multiple independent events occur simultaneously, like finding the probability that both a coin lands on heads and a die lands on three.

Calculating Independent Probability

To calculate independent probability, it's all about using the multiplication rule with the given probabilities of the independent events. For example, let's find the probability of both event Q and event R occurring if they are independent.

Given that:

• \( P(Q) = \frac{1}{4} \)
• \( P(R) = \frac{2}{3} \)

We apply the multiplication rule:
The probability of both Q and R happening together is:
\[ P(Q \text{ and } R) = P(Q) \cdot P(R) = \frac{1}{4} \times \frac{2}{3} \]

This multiplication gives us:\[ P(Q \text{ and } R) = \frac{1}{6} \]

Thus, there's a 1 in 6 chance that both events will occur simultaneously.