More Trials

Re-did the QASM test with our code for a 50% probability, only upped the number of runs

from 1024 to 4096. The results were indeed better: we got 2075 successes out of 4096, thus .507

where we were getting 530 out of 2024 for .517.

                                                                                 

Real:

                                                                         

 

Ideal:

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Gemini 1.5 Pro

Binomial Dsitribution; help me understand why, when I run more trials, my results are more accurate but -at the same time - the probability of my actual result is lower…

When dealing with a binomial distribution, understanding the role of increasing the number of trials helps make sense of various results and their probabilities.

1. Law of Large Numbers: As you run more trials, the experimental or empirical results tend to get closer to the theoretical or expected value of the distribution. This is due to the Law of Large Numbers, which states that as the number of trials increases, the sample mean will converge to the expected value. This results in more accurate results in terms of capturing the true probability of success.

2. Probability of Specific Outcomes: As the number of trials increases, the binomial distribution becomes more spread out. With a larger number of trials, the variance increases, and the probabilities are distributed across more possible outcomes. This means that any specific result, particularly one that occurs at the tails of the distribution, will have a lower probability. In simpler terms, while the outcome you're observing might be more accurate in reflecting the underlying probability, the chance of observing exactly that outcome in a vast number of possibilities becomes smaller.

3. Central Limit Theorem: Additionally, as the number of trials increases, the binomial distribution tends to resemble a normal distribution, especially when the probability of success is not too close to 0 or 1. The larger spread of possible outcomes further reduces the probability of landing on any single specific number of successes.

In summary, more trials lead to results that more closely reflect the expected probability, but they simultaneously result in a lower probability for any specific number of successes due to the broader distribution of possible outcomes. 

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Gemini:

What is the Central Limit Theorem, and what is it used for?

The Central Limit Theorem (CLT) is a fundamental concept in statistics that states:

"The distribution of sample means approaches a normal distribution as the sample size gets larger, regardless of the shape of the population distribution."

Here's a breakdown of what this means:

Sample Means: Imagine taking multiple samples from a population (e.g., measuring the heights of 30 people, then another 30, then another 30, and so on).  Each sample will have its own average (mean). 

Distribution of Sample Means: If you plot all these sample means, you'll get a distribution of means. 

Normal Distribution: The CLT says that as your sample sizes get bigger, this distribution of sample means will start to look more and more like a bell-shaped curve, which is known as a normal distribution. 

Why is this important?

Inference: The CLT allows us to make inferences about a population based on a sample. Even if the original population data isn't normally distributed, we can use the CLT to assume that the distribution of sample means is approximately normal. This is crucial for hypothesis testing and confidence intervals.

Estimation: The CLT helps us estimate population parameters (like the population mean) using sample statistics.

Simplification: The normal distribution is well-understood and has many convenient properties. The CLT allows us to use these properties even when dealing with data that isn't normally distributed.

How is it used?

Here are some common applications of the Central Limit Theorem:

Quality Control:  Companies use the CLT to ensure that products meet certain quality standards. They take samples of products and use the CLT to determine if the average quality of the entire production run is within acceptable limits.

Medical Research:  Researchers use the CLT to analyze data from clinical trials. They can use the CLT to determine if a new treatment is effective by comparing the average outcomes of patients who received the treatment to those who received a placebo.

Social Sciences:  Social scientists use the CLT to analyze survey data. They can use the CLT to determine if there is a difference in the opinions of two groups of people.

Finance:  Financial analysts use the CLT to model the behavior of stock prices. They can use the CLT to estimate the probability of a stock price going up or down.

In summary:

The Central Limit Theorem is a powerful tool that allows us to use the properties of the normal distribution to analyze data, even when the original data isn't normally distributed. This makes it a cornerstone of statistical inference and decision-making in various fields.