Calculating Binomial Probability with Normal Approximation

In this article, we will explore how to calculate the probability of achieving a specific number of successes in a series of trials using the binomial distribution, and how to simplify this process with the normal approximation. We will focus on the case of 25 trials where the probability of success for each trial is 0.44, and specifically calculate the probability of exactly 15 successes.

Understanding the Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a series of independent trials, each with the same probability of success. The probability mass function for a binomial distribution is given by the formula:

[P(X k) binom{n}{k} p^k (1-p)^{n-k}]

Where:

n is the number of trials k is the number of successes p is the probability of success on a single trial binom{n}{k} is the binomial coefficient, calculated as frac{n!}{k!(n-k)!}

Exact Calculation Using Binomial Distribution

In our specific example, we have n 25, k 15, and p 0.44. First, we need to calculate the binomial coefficient:

[binom{25}{15} frac{25!}{15!10!} approx 3,268,760]

Next, we calculate the powers:

[0.44^{15} approx 1.017 times 10^{-5}] [0.56^{10} approx 0.0064]

Finally, we plug these values into the binomial probability formula:

[P(X 15) 3,268,760 times 1.017 times 10^{-5} times 0.0064 approx 0.207]

Therefore, the probability of getting exactly 15 successes in 25 trials with a success probability of 0.44 is approximately 0.207 or 20.7%.

Using Normal Approximation for Simplicity

The binomial distribution can be approximated using the normal distribution when the number of trials is large and the probability of success is not too close to 0 or 1. This simplifies the calculation process significantly.

Calculating Parameters of the Normal Distribution

First, we calculate the mean (μ) and standard deviation (σ) of the binomial distribution:

[mu np 25times 0.44 11] [sigma sqrt{np(1-p)} sqrt{11times 0.56} sqrt{6.16} approx 2.5]

Using these parameters, we can approximate the binomial distribution with a normal distribution and calculate the probability using the cumulative distribution function (CDF).

Applying the Normal Approximation

Using the normal approximation, we calculate:

[P(X 15) approx P(14.5 leq X leq 15.5)]

This translates to:

[Pleft( frac{15 - 11}{2.5} leq Z leq frac{15.5 - 11}{2.5} right) P(1.6 leq Z leq 1.81310)]

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

[P(Z leq 1.81310) approx 0.965092] [P(Z leq 1.41019) approx 0.920758]

Therefore, the probability is:

[P(15 leq X leq 15.5) 0.965092 - 0.920758 0.0443]

This is very close to the value obtained using the binomial distribution method.

Conclusion

In conclusion, both the exact binomial calculation and the normal approximation provide a meaningful way to estimate the probability of achieving a specific number of successes. The normal approximation offers a simpler method, especially for large numbers of trials. Understanding these methods can greatly enhance your ability to solve complex probability problems efficiently.