Problem 30 - Calculus for Biology and Medicine

Question

\(S_{n}\) is binomially distributed with parameters \(n\) and \(p\). 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

Exact: \(0.1123\), Poisson: \(0.1118\), Normal: \(0.1129\).

Step 1: Use the Binomial Formula

To find the exact probability, we use the binomial probability formula: \(P(S_n = k) = \binom{n}{k} p^k (1-p)^{n-k}\). For this exercise, plug in \(n=50\), \(k=25\), and \(p=0.5\). Calculate \(\binom{50}{25}\), which is the number of combinations of 50 items taken 25 at a time.

Step 2: Calculating the Exact Probability

Calculate \(P(S_{50} = 25) = \binom{50}{25} (0.5)^{25} (0.5)^{25} \). This simplifies to \(\binom{50}{25} (0.5)^{50} \). Computing this gives the exact probability.

Step 3: Poisson Approximation

To use a Poisson approximation, note the parameters are large enough that the Binomial distribution can be approximated by a Poisson distribution when \(n\) is large. The mean \(np\) should be used as the Poisson parameter \(\lambda\), giving us \(\lambda = 50 \times 0.5 = 25\). Calculate \(P(S_{n} = 25) \approx \frac{\lambda^{25} e^{-\lambda}}{25!} \).

Step 4: Normal Approximation

For the normal approximation, we treat \(S_{n}\) as a normal distribution with mean \(np = 25\) and variance \(np(1-p) = 12.5\). Calculate \(P(24.5 < S_n < 25.5)\) using the standard normal distribution: \(P(-0.14 < Z < 0.14)\), where \(Z\) is the standard normal variable. Use a Z-table or calculator to find this probability.

Key Concepts

Poisson Approximation

The Poisson approximation is a useful tool when dealing with binomial distributions where the number of trials, \(n\), is large. This approximation is based on the idea that when \(n\) is large and the probability of success, \(p\), is small, the binomial distribution approaches the Poisson distribution. The key is to set the Poisson parameter, \( \lambda \), equal to the expected value of the binomial distribution, which is \( np \).

For our exercise, \(n = 50\) and \(p = 0.5\), so \( \lambda = 50 \times 0.5 = 25\). The formula for a Poisson probability is:

• \( P(S_n = k) \approx \frac{\lambda^k e^{-\lambda}}{k!} \)

Plugging in \(k = 25\), we calculate the approximate probability:

• \( P(S_n = 25) \approx \frac{25^{25} e^{-25}}{25!} \)

This method provides a quick approximation when conditions fit, making it a powerful alternative to direct binomial calculations.

Normal Approximation

The normal approximation to a binomial distribution is based on the Central Limit Theorem, which states that the distribution of the sample mean will tend to a normal distribution as the sample size becomes large. This approximation requires that both \(np\) and \(n(1-p)\) are greater than 5, ensuring the distribution is not highly skewed.

For a binomial distribution \(S_n)\) with \(n = 50\) and \(p = 0.5\), we find:

• Mean, \( \mu = np = 25 \)
• Variance, \( \sigma^2 = np(1-p) = 12.5 \)
• Standard deviation, \( \sigma = \sqrt{12.5} \approx 3.54 \)

To find the probability \( P(24.5 < S_n < 25.5) \), we standardize the values to obtain \( Z \) scores:

• \( Z = \frac{X - \mu}{\sigma} \)
• For 24.5 and 25.5, \( Z \approx -0.14 \) and \( Z \approx 0.14 \) respectively

This results in calculating \( P(-0.14 < Z < 0.14) \), which can be found using a Z-table.

Binomial Formula

The binomial formula is the core for calculating exact probabilities in a binomial distribution. This distribution arises when there are a fixed number of trials, each trial has two possible outcomes, and each trial is independent. The formula for these probabilities is:

• \( P(S_n = k) = \binom{n}{k} p^k (1-p)^{n-k} \)

For our problem, the parameters are \(n = 50\), \(k = 25\), and \(p = 0.5\).

To compute the exact probability:

• The combination \( \binom{n}{k} \) represents the number of ways to choose \(k\) successes from \(n\) trials.
• In our case, \( \binom{50}{25} \) calculates all possible combinations of 25 successes in 50 trials.
• The probability for each specific combination is \( (0.5)^{25} (0.5)^{25} \), simplifying to \( (0.5)^{50} \).

Thus, the exact probability is found by multiplying these components for a complete picture of likelihood.

Probability Calculation

Calculating probabilities for different types of distributions involves using specific formulas that account for the parameters of each distribution.

When approaching a binomial distribution:

• We use the binomial formula: \( P(S_n = k) = \binom{n}{k} p^k (1-p)^{n-k} \) to find exact probabilities.
• This requires determining the combinations and powers of probabilities for successes and failures.

For a Poisson approximation:

• The formula \( \frac{\lambda^k e^{-\lambda}}{k!} \) depends on the expected successes \( \lambda \).

For the normal approximation:

• We calculate \( Z \)-scores and use the standard normal distribution to find the probability.
• This approach makes use of known properties of the normal distribution, such as the cumulative distribution function.

Understanding these methods enhances problem-solving skills and ensures accurate estimation of probabilities in statistical contexts.