Problem 30 - Calculus for Biology and Medicine

Question

For \(n=50\) and \(p=0.5\), compute \(P\left(S_{n}=25\right)\) (a) exactly, (b) by using a Poisson approximation, and (c) by using a normal approximation.

Step-by-Step Solution

Answer

The exact probability is 0.1123; Poisson approx. gives 0.0796, and normal approx. gives 0.1115.

Step 1: Understanding the Problem

The problem asks us to find the probability \(P(S_n = 25)\) for a binomial random variable with \(n=50\) and \(p=0.5\). This is the probability of having exactly 25 successes in 50 trials, where each trial has a success probability of 0.5.

Step 2: Exact Calculation Using Binomial Formula

The exact probability is computed using the binomial probability formula: \[ P(S_n = 25) = \binom{50}{25} (0.5)^{25} (0.5)^{25} \]Calculate \(\binom{50}{25}\), which is the number of ways to choose 25 successes out of 50 trials, then multiply by \((0.5)^{50}\). \[ \binom{50}{25} = \frac{50!}{25!25!} = 126410606437752 \]So, \( P(S_n = 25) = \frac{126410606437752}{1125899906842624} \approx 0.112275 \).

Step 3: Poisson Approximation

The Poisson approximation can be used when \( n \) is large and \( p \) is small. The parameter for the Poisson distribution \( \lambda = np = 50 \times 0.5 = 25 \). From Poisson probability mass function:\[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \]Substitute the values:\[ P(X = 25) \approx \frac{e^{-25} 25^{25}}{25!} \approx 0.0795892 \].

Step 4: Normal Approximation

With a large \(n\), the binomial distribution can be approximated by a normal distribution: \[ S_n \sim N(np, np(1-p)) = N(25, 12.5) \]. Use the continuity correction to improve accuracy: calculate \[ P(24.5 < X < 25.5) \].Convert to the standard normal distribution:\[ Z_1 = \frac{24.5 - 25}{\sqrt{12.5}} \approx -0.14 \]\[ Z_2 = \frac{25.5 - 25}{\sqrt{12.5}} \approx 0.14 \]Using the standard normal table, find \[ P(24.5 < X < 25.5) \approx P(-0.14 < Z < 0.14) \approx 0.1115 \].

Key Concepts

Poisson Approximation

When dealing with a large number of trials and a small probability of success, the Poisson approximation is a handy tool to estimate binomial probabilities. In the context of our exercise, the Poisson approximation makes sense since we have 50 trials, although the success probability (0.5) is not small, which is a slight deviation from the ideal use case. Here's how it works:

• You calculate the parameter \(\lambda\) as the product of the number of trials \(n\) and the probability \(p\). For this exercise, \(\lambda = np = 50 \times 0.5 = 25\).
• Using the Poisson probability mass function, you estimate the probability: \[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}\]
• Substitute \(k = 25\) to find: \[ P(X = 25) \approx \frac{e^{-25} 25^{25}}{25!} \approx 0.0795892 \]

Remember, the Poisson approximation suits better when \(p\) is small and \(n\) is large, aiming to simplify calculations significantly by avoiding cumbersome computations of binomial coefficients.

Normal Approximation

Normal approximation is used for approximating binomial distributions when the number of trials \(n\) is large. It's based on the Central Limit Theorem, which essentially tells us that as the sample size increases, the binomial distribution approaches a normal distribution. Here's why it's useful:

• For large \(n\), calculating exact probabilities using the binomial formula becomes complex, so we pick normal approximation as a more effortless approach.
• You first identify the parameters of the normal distribution: the mean \(\mu = np\) and the variance \(\sigma^2 = np(1-p)\). In our problem, \(\mu = 25\) and \(\sigma^2 = 12.5\).
• The next step is to convert the specific problem into the standard normal variable \(Z\) using these parameters.

Due to the approximative nature, this approach is ideal for scenarios where neither \(p\) is too high nor too low, and becomes particularly effective after applying a continuity correction, which we'll discuss next.

Continuity Correction

To get better accuracy when using normal approximation for discrete distributions, we apply the continuity correction. This is particularly crucial because the normal distribution is continuous, while the binomial distribution is discrete.Consider the continuity correction as a fine-tuning step to bridge the small gap between the discrete and continuous nature:

• It involves adjusting the values of \(X\) by \(0.5\) since the binomial distribution is about exact counts.
• For the problem \(P(S_n = 25)\), you actually compute it as \(P(24.5 < X < 25.5)\) in the continuous realm.
• Convert these bounds into \(Z\)-scores: \[ Z_1 = \frac{24.5 - 25}{\sqrt{12.5}} \approx -0.14 \] \[ Z_2 = \frac{25.5 - 25}{\sqrt{12.5}} \approx 0.14 \]
• Check standard normal distribution tables or use software to find \[P(-0.14 < Z < 0.14) \approx 0.1115\]

This continuity adjustment helps in approximating binomial probabilities more accurately with the normal distribution model, improving the reliability of the results.