Approximate The Distribution With A Normal Distribution, N ( 14 , 2.6 N(14, 2.6 N ( 14 , 2.6 ], Where The Mean ( Μ \mu Μ ) Is 14 And The Standard Deviation ( Σ \sigma Σ ) Is 2.6.Using This Approximation, Find The Probability Of Flipping 18 Or 19 Heads. You

Introduction

In probability theory, approximating a distribution with a normal distribution is a common technique used to simplify complex problems. In this article, we will explore how to approximate the distribution of flipping a coin with a normal distribution, $N(14, 2.6)$, where the mean ($\mu$) is 14 and the standard deviation ($\sigma$) is 2.6. We will then use this approximation to find the probability of flipping 18 or 19 heads.

Understanding the Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In the case of the normal distribution $N(14, 2.6)$, the mean is 14 and the standard deviation is 2.6.

Properties of the Normal Distribution

The normal distribution has several important properties that make it a useful tool for approximating other distributions. Some of the key properties of the normal distribution include:

• Symmetry: The normal distribution is symmetric about the mean, which means that the probability of data being above the mean is equal to the probability of data being below the mean.
• Bell-shaped curve: The normal distribution has a bell-shaped curve, which means that the probability of data being close to the mean is higher than the probability of data being far from the mean.
• Mean and standard deviation: The mean and standard deviation of the normal distribution are the parameters that define the distribution.

Approximating the Distribution with a Normal Distribution

To approximate the distribution of flipping a coin with a normal distribution, we need to find the mean and standard deviation of the distribution. In this case, the mean is 14 and the standard deviation is 2.6.

Calculating the Probability of Flipping 18 or 19 Heads

To calculate the probability of flipping 18 or 19 heads, we need to use the normal distribution to approximate the distribution of the number of heads. We can do this by using the following formula:

P(X ≥ 18) = P(Z ≥ (18 - 14) / 2.6) P(X ≥ 19) = P(Z ≥ (19 - 14) / 2.6)

where Z is the standard normal distribution.

Using the Standard Normal Distribution

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. We can use the standard normal distribution to calculate the probability of flipping 18 or 19 heads.

Calculating the Probability

To calculate the probability of flipping 18 or 19 heads, we need to use the standard normal distribution to find the probability of Z being greater than or equal to the given value.

P(Z ≥ (18 - 14) / 2.6) = P(Z ≥ 1.54) P(Z ≥ (19 - 14) / 2.6) = P(Z ≥ 1.73)

Using a standard normal distribution table or calculator, we can find the following probabilities:

P(Z ≥ 1.54) = 0.0611 P(Z ≥ 1.73) = 0.0414

Finding the Final Probability

To find the final probability of flipping 18 or 19 heads, we need to add the probabilities of flipping 18 heads and flipping 19 heads.

P(X ≥ 18) = P(Z ≥ (18 - 14) / 2.6) = 0.0611 P(X ≥ 19) = P(Z ≥ (19 - 14) / 2.6) = 0.0414

P(X ≥ 18 or 19) = P(X ≥ 18) + P(X ≥ 19) = 0.0611 + 0.0414 = 0.1025

Conclusion

In this article, we approximated the distribution of flipping a coin with a normal distribution, $N(14, 2.6)$, and used this approximation to find the probability of flipping 18 or 19 heads. We calculated the probability of flipping 18 or 19 heads using the standard normal distribution and found that the probability is approximately 0.1025.

References

• Normal Distribution: A probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.
• Standard Normal Distribution: A normal distribution with a mean of 0 and a standard deviation of 1.
• Probability: A measure of the likelihood of an event occurring.

Future Work

In the future, we can use this approximation to find the probability of flipping other numbers of heads. We can also use this approximation to find the probability of other events, such as the probability of flipping a certain number of tails.

Code

import numpy as np
mean = 14
std_dev = 2.6

num_heads = 18

prob = 1 - np.cumsum(np.exp(-((np.arange(num_heads, 100) - mean) / std_dev)**2))[-1]
print("The probability of flipping", num_heads, "heads is:", prob)

num_heads = 19

prob = 1 - np.cumsum(np.exp(-((np.arange(num_heads, 100) - mean) / std_dev)**2))[-1]
print("The probability of flipping", num_heads, "heads is:", prob)

This code uses the numpy library to calculate the probability of flipping a certain number of heads. It defines the mean and standard deviation of the normal distribution, and then calculates the probability of flipping a certain number of heads using the cumulative distribution function of the normal distribution.