Problem 28 - Calculus for Biology and Medicine

Question

\(S_{n}\) is binomially distributed with parameters \(n\) and \(p\). For \(n=100\) and \(p=0.1\), compute \(P\left(S_{n}=10\right)\) (a) exactly, (b) by using a Poisson approximation, and (c) by using a normal approximation.

Step-by-Step Solution

Answer

Exact: Use binomial formula; Poisson: Approximate with \( \lambda = 10 \); Normal: Use \( Z \) with continuity correction.

Step 1: Understand the Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, each with success probability \( p \). In this problem, \( S_n \) follows a binomial distribution with parameters \( n = 100 \) and \( p = 0.1 \).

Step 2: Exact Binomial Calculation

To find \( P(S_n = 10) \) exactly, use the binomial probability formula: \[P(S_n = k) = \binom{n}{k} p^k (1-p)^{n-k}\]Plug in \( n = 100 \), \( k = 10 \), and \( p = 0.1 \):\[P(S_n = 10) = \binom{100}{10} (0.1)^{10} (0.9)^{90}\]Calculate \( \binom{100}{10} = \frac{100!}{10!(100-10)!} \) and then substitute the values into the equation to find the probability.

Step 3: Poisson Approximation

When \( n \) is large and \( p \) is small, the binomial distribution can be approximated by a Poisson distribution with parameter \( \lambda = np = 10 \). The Poisson probability is given by:\[P(S_n = k) = \frac{e^{-\lambda} \lambda^k}{k!}\]Substitute \( \lambda = 10 \) and \( k = 10 \) to find:\[P(S_n = 10) = \frac{e^{-10} \cdot 10^{10}}{10!}\]

Step 4: Normal Approximation

The binomial distribution can also be approximated by a normal distribution with mean \( \mu = np = 10 \) and variance \( \sigma^2 = np(1-p) = 9 \). Use the continuity correction and standard normal distribution:\[P(9.5 < S_n < 10.5) \approx P\left( \frac{9.5-10}{3} < Z < \frac{10.5-10}{3} \right)\]This simplifies to:\[P(-0.1667 < Z < 0.1667)\]Find this probability using the standard normal distribution tables or software.

Key Concepts

Poisson Approximation

When dealing with a binomial distribution where the number of trials is quite large (like our example with 100 trials) and the probability of success is small (in this case, 0.1), the distribution can be well-approximated by a Poisson distribution. The essence here is that instead of working with the cumbersome binomial calculations, the Poisson distribution provides a simpler formula for probability.
The parameter \( \lambda \) for the Poisson approximation is derived from the product of the number of trials \( n \) and the probability of success \( p \). Therefore, \( \lambda = np = 10 \). The Poisson probability mass function is expressed as:

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

This formula allows you to calculate probabilities easily when the conditions for approximation are met. By substituting \( \lambda = 10 \) and \( k = 10 \) into the formula, you can find the approximate probability, which offers a quick and efficient way to estimate the binomial probability without calculating large binomial coefficients.

Normal Approximation

Another useful approximation for a binomial distribution is the normal approximation. It becomes effective when both \( n \) is large and neither \( p \) nor \( 1-p \) is too close to 0 or 1. In our example, since \( n = 100 \) and \( p = 0.1 \), the conditions are sufficiently satisfied. A normal distribution can approximate the binomial distribution with a mean \( \mu = np \) and a variance \( \sigma^2 = np(1-p) \).
For our specific problem, \( \mu = 10 \) and \( \sigma^2 = 9 \), or \( \sigma = 3 \). Since we're approximating a discrete distribution with a continuous one, it's important to apply a continuity correction. This means adjusting your target value by 0.5.
To find \( P(S_n = 10) \), compute:

• \( P(9.5 < S_n < 10.5) \approx P\left( \frac{9.5-10}{3} < Z < \frac{10.5-10}{3} \right) \)
• This simplifies to \( P(-0.1667 < Z < 0.1667) \)

After determining the Z-scores, use standard normal distribution tables or software to find the probability, offering a straightforward way to approximate when exact binomial calculations are impractical.

Probability Calculation

Calculating probabilities directly from distributions is crucial in understanding how likely certain outcomes are. To tackle probabilities in binomial distribution correctly, start with the binomial probability formula:

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

This formula accounts for permutations (or combinations) of success and failure across your trials.
For a very concrete example, to find exact probability \( P(S_{n} = 10) \), you would calculate:

• \( \binom{100}{10} = \frac{100!}{10!(100-10)!} \)
• Then, substitute and solve using \( p = 0.1 \) and \( 1-p=0.9 \).

Sometimes, these exact calculations can be extensive as you'll deal with large factorials. Therefore, approximations like Poisson and Normal may come in handy as they simplify the computation using their respective formulas to deliver similar probabilistic insights with much less computational effort.