Problem 16 - Calculus for Biology and Medicine

Question

Toss a fair coin 150 times. Use the central limit theorem and the histogram correction to find an approximation for the probability that the number of heads is at least 70 .

Step-by-Step Solution

Answer

The probability that the number of heads is at least 70 is approximately 0.8159.

Step 1: Understand the parameters of the problem

We are tossing a fair coin 150 times. A fair coin means that the probability of getting heads, \( p \), is 0.5. The number of trials, \( n \), is 150. We are interested in the probability of getting at least 70 heads.

Step 2: Define the random variable

Let \( X \) be the random variable representing the number of heads in 150 tosses. Since each coin toss is independent and has identical distribution, \( X \) follows a binomial distribution: \( X \sim \text{Binomial}(n=150, p=0.5) \).

Step 3: Apply Central Limit Theorem (CLT)

According to the Central Limit Theorem, for a large \( n \), the binomial distribution can be approximated by a normal distribution: \( X \sim N(\mu, \sigma^2) \), where \( \mu = np = 150 \times 0.5 = 75 \) and \( \sigma = \sqrt{np(1-p)} = \sqrt{150 \times 0.5 \times 0.5} = \sqrt{37.5} \).

Step 4: Apply the continuity correction

Since we are using a normal approximation to a discrete distribution, apply a continuity correction. We seek \( P(X \geq 70) \). The continuity correction implies we should find \( P(X > 69.5) \).

Step 5: Convert to standard normal distribution

Standardize using \( Z = \frac{X - \mu}{\sigma} \). Substituting the values: \( Z = \frac{69.5 - 75}{\sqrt{37.5}} \approx \frac{-5.5}{6.12} \approx -0.899 \).

Step 6: Find the probability using the standard normal distribution

Using the standard normal distribution table, \( P(Z > -0.899) \) corresponds to \( 1 - P(Z < -0.899) \), which is approximately 0.8159.

Key Concepts

Binomial Distribution

A binomial distribution is a discrete probability distribution that represents the number of successes in a fixed number of independent trials, each with the same probability of success. In this context, when tossing a fair coin 150 times, each coin toss represents a trial. The probability of landing a head (considered a success) is 0.5 for a fair coin.

Therefore, in this scenario, the situation is modeled as a binomial distribution because the trials have fixed probabilities and are independent of each other. We write this as \( X \sim \text{Binomial}(n=150, p=0.5) \), where \( X \) is the random variable representing the number of heads obtained.

Normal Approximation

The normal approximation to the binomial distribution involves approximating the binomial distribution by a normal distribution when the number of trials is large. According to the Central Limit Theorem, if the number of trials \( n \) of a binomial distribution is sufficiently large, the distribution of the sample mean will approximate a normal distribution, regardless of the original distribution of the data.

For our problem, with 150 trials, the mean \( \mu \) of the normal approximation is equal to \( np = 75 \) and the variance \( \sigma^2 \) is \( np(1-p) = 37.5 \). The normal approximation simplifies our calculations, allowing us to use techniques from the normal distribution, such as the standard normal tables, to find probabilities.

Continuity Correction

When approximating a discrete distribution, like the binomial, by a continuous distribution, such as the normal distribution, a continuity correction is used. This correction accounts for the differences between discrete and continuous distributions. It slightly shifts the boundaries of our discrete data intervals.

For example, when calculating the probability of obtaining at least 70 heads, we initially calculate \( P(X \geq 70) \). With continuity correction, we adjust this to \( P(X > 69.5) \). This adjustment allows for more accurate approximation and interpretation of the probability value.

Random Variable

A random variable is a fundamental concept in probability and statistics, representing a variable that can take on different values based on the outcomes of a random phenomenon. In our problem, the random variable \( X \) represents the number of heads obtained from tossing a fair coin 150 times.

Since each toss is independent, \( X \) is modeled by a binomial distribution, which comes from a series of independent Bernoulli trials. The choice of \( X \), as the random variable indicating the number of heads, is crucial for setting up our probability model and determining how we use statistical methods like the normal approximation and continuity correction for analysis.