Question

Use Bernoulli's formula to show that if greater certainty is wanted in the problem of Exercise 7 , say, \(c=10,000\), then the number of trials necessary is \(N=31,258\).

Short answer

Answer: The minimum number of trials required to achieve a certainty of 10,000 is 31,258 trials.

Step-by-step solution

Understand Bernoulli's Inequality

Bernoulli's inequality states that for any positive integer \(k\) and any real number \(x ≥ -1\), we have \((1+x)^k ≥ 1+kx\). In our case, we will use this inequality to find the number of trials necessary for achieving a specific level of certainty.

Apply Bernoulli's Inequality to the Problem

Let \(p\) be the probability of success in each trial, and let \(q = 1-p\) be the probability of failure. We want to find the number of trials \(N\) such that the certainty, \(c\), is \(10,000\). This implies that we want the probability of having at least one success to be at least \(\frac{10,000}{10,000+1}\). We know that the probability of having no successes in \(N\) trials is \(q^N\), so the probability of having at least one success is \(1-q^N\). We want to find the smallest \(N\) such that: \(1-q^N ≥ \frac{10,000}{10,000+1}\) Now use Bernoulli's Inequality with \(k=N\), \(x=-q\), and consider that the worst possible case is when \(p=q=\frac{1}{2}\): \((1+(-q))^N = (1-q)^N≥ 1-Nq\)

Find the Smallest N Satisfying the Inequality

Since we want the certainty to be at least \(\frac{10,000}{10,001}\), we want to find the smallest \(N\) such that: \(1-Nq ≥ \frac{10,000}{10,000+1}\) Now, remember that we have considered the worst possible case when \(p=q=\frac{1}{2}\), so \(1-Nq =\frac{1}{2} -\frac{N}{2}\) Thus, the inequality becomes: \(\frac{1}{2} -\frac{N}{2} ≥ \frac{10,000}{10,000+1}\) Now, we solve for N: \(N ≤ 2\cdot\frac{10,000}{1-\frac{10,000}{10,000+1}}\) After calculating this expression, we get: \(N ≤ 31,257.94\) Since the number of trials must be an integer, we take the smallest integer greater than this value, which is: \(N = 31,258\) Thus, the number of trials necessary to achieve a certainty of 10,000 in the problem is \(N=31,258\).

Key concepts

Probability of Success

The concept of probability of success is crucial in determining the outcome of events in trials, especially when using Bernoulli's formula. In any trial, the probability of success (denoted as \( p \)) represents the likelihood that a desired event will occur. This number must be between 0 and 1, where 1 indicates certain success, and 0 indicates no chance of success.

When applying Bernoulli's inequality to find the number of trials necessary for a certain event to happen with a specific confidence level, understanding \( p \) is important. For example, in some cases, such as when maximum uncertainty is considered, \( p \) might equal 0.5, showing equal chances of success or failure. This is often referred to as the worst-case scenario because it assumes the least probable favorable outcome for achieving the desired certainty.

• Probability of success (\( p \)): Likelihood of a successful event in a single trial.
• Ranges between 0 and 1.
• For challenging scenarios, assume \( p = 0.5 \) for calculations.

By knowing \( p \), one can assess how trials are likely to develop and formulate strategies to adjust the number of trials needed to achieve desired certainty levels.

Number of Trials

The number of trials \( N \) refers to the repetitions of the event needed to achieve a particular level of certainty. In probabilistic terms, more trials often lead to higher certainty that a certain number of successes will occur. To determine this effectively, Bernoulli's inequality can be applied.

In the exercise, we needed to find \( N \) such that at least one success occurs with high certainty. The problem states that a certainty of 10,000 out of 10,001 is required, which translates into a very high probability of success over multiple trials. Calculating \( N \) involves considering the worst-case scenario with minimal certainty of individual successes.

• Number of Trials (\( N \)): The required count of event repetitions for a targeted certainty.
• Higher values of \( N \) increase the likelihood of achieving higher confidence levels.
• Use Bernoulli's Inequality to compute \( N \) for specific certainty levels.

This systematic approach allows one to decide the necessary number of trials proactively, ensuring that the certainty and success of outcomes meet specified probabilities.

Level of Certainty

The level of certainty refers to the degree of confidence we aim to achieve regarding the probability of a successful outcome. In probability and statistics, higher certainty demands more trials, assuming a constant probability of success per trial. This requirement often leads to a trade-off between the effort (more trials) and the assurance of success.

In the given problem, the level of certainty was set at 10,000 to 10,001, which means the probability of achieving at least one success was immensely high. To reach such a confidence level, setting the probability of failure (\( q \), where \( q = 1 - p \)) to a sufficiently low value was necessary by increasing the number of trials through Bernoulli's formulation.

• Level of Certainty: The desired confidence in achieving at least one success.
• Higher certainty typically requires more trials, assuming a given probability of success.
• Calculated such that probability of success meets the desired thresholds.

Achieving high levels of certainty ensures that the outcomes of trials align closely with expectations, enhancing predictive accuracy in probabilistic examinations.