91Ó°ÊÓ

In a binomial distribution, the "probability of success" refers to the likelihood that a single trial results in success. Each trial is independent, and the probability of success remains constant across all trials. In the context of the problem, we are dealing with a binomial distribution where the probability of success in each trial, denoted by \( p \), is \( \frac{1}{4} \), or 0.25.
This means that each trial has a 25% chance of being successful, while there's a 75% chance of failure (\( 1-p = \frac{3}{4} \)).
In the exercise, the task is to ensure that having at least one success across multiple trials (\( n \) trials) is highly probable—specifically, with a probability of at least \( \frac{9}{10} \) or 90%.

• Success is defined per the context: for example, rolling a 6 on a die, flipping a heads on a coin, etc.
• In scenarios like these, we apply the binomial formula or logic to determine the outcomes we desire based on given parameters.

This foundation lays the groundwork for understanding the probability of success in any given trial series, essential for advancing in binomial analyses.

In this exercise, we are tasked with solving an inequality that ensures the probability condition is met. We derived the inequality:
\[ (\frac{3}{4})^n \leq \frac{1}{10} \]
This inequality needs to be solved to find the smallest integer value of \( n \) such that the inequality holds true.

• We start by expressing the desired criterion using an important binomial probability property. The inequality transformation stems from recognizing that \( 1 - (1-p)^n \) represents the probability of at least one success.
• Performing algebraic manipulations allows us to isolate \( n \). This transformation is pivotal for translating probability into a more manageable form involving power relations.

Solving such inequalities is a crucial skill. It bridges the context-specific problem settings with algebraic methods, thus leading to viable solutions.

Logarithms are incredibly useful in solving equations and inequalities involving exponentials, like our equation \((\frac{3}{4})^n \leq \frac{1}{10}\). By taking the logarithm of both sides, we can eliminate the exponential form and solve for \( n \).
Applying logarithms here, our inequality becomes:
\[ n \cdot \log_{10}(\frac{3}{4}) \leq \log_{10}(0.1) \]
Logarithms help us to express the powers in terms of products, allowing for easier solution processes. Here’s a step-by-step breakdown:

• Take the log of each side regarding equality: This strategy helps when we need to bring down the power of \( n \) using the properties of logarithms, as \( n \cdot \log(b) \).
• Apply the rule \( \log(a^b) = b \cdot \log(a) \): This is crucial for turning exponential expressions into linear form.
• Simplify the expression: Once logged, rearrange the equation to build one where \( n \) is clearly isolated.

Through this, we simplify and understand better how values in exponential problems like these can align to a single value or set of solutions. Understanding logarithms will aid in cracking similar problems with ease.