Problem 15 - Calculus for Biology and Medicine

Question

15\. Toss a fair coin 400 times. Use the central limit theorem and the histogram correction to find an approximation for the probability of getting at most 190 heads.

Step-by-Step Solution

Answer

The probability of getting at most 190 heads is approximately 0.1711.

Step 1: Define the Problem

We need to determine the probability of getting at most 190 heads in 400 coin tosses of a fair coin. A fair coin has two outcomes, heads or tails, each with a probability of 0.5.

Step 2: Set up Parameters for Binomial Distribution

The number of heads in 400 tosses follows a binomial distribution with parameters \(n = 400\) and \(p = 0.5\). The mean \(\mu\) and standard deviation \(\sigma\) of this distribution are given by \(\mu = np\) and \(\sigma = \sqrt{np(1-p)}\).

Step 3: Calculate the Mean and Standard Deviation

Calculate \(\mu = 400 \times 0.5 = 200\) and \(\sigma = \sqrt{400 \times 0.5 \times 0.5} = 10\).

Step 4: Apply the Central Limit Theorem

The central limit theorem states that for a large number of trials, the distribution of the sample mean will approximate a normal distribution. Therefore, we can treat the number of heads as a normal random variable with mean 200 and standard deviation 10.

Step 5: Standardize the Variable

To find the probability of getting at most 190 heads, convert 190 to the standard normal variable: \(z = \frac{190 - 200}{10} = -1.0\).

Step 6: Use the Normal Distribution Table

Using the standard normal distribution table, find the probability corresponding to \(z = -1.0\). The approximate probability is 0.1587.

Step 7: Incorporate the Histogram Correction

To include the histogram correction (continuity correction), consider \(P(X \leq 190.5)\). Calculate the corrected \(z\)-value: \(z = \frac{190.5 - 200}{10} = -0.95\).

Step 8: Find the Corrected Probability

Using the z-table, the probability for \(z = -0.95\) is approximately 0.1711.

Key Concepts

Binomial Distribution

The binomial distribution is a fundamental concept in probability theory. It describes the number of successes in a fixed number of independent trials, with each trial having two possible outcomes: success or failure. In the case of a fair coin toss, the outcomes are either heads or tails.

• Each coin toss in our exercise is a trial, where getting a head is considered a success with probability, \( p = 0.5 \).
• The total number of trials, \( n \), is 400, representing the total number of coin tosses.

The binomial distribution's parameters are derived from these probabilities: the mean \( \mu \) is computed as \( np \), and the standard deviation \( \sigma \) is \( \sqrt{np(1-p)} \).

In our case, the calculation gives us a mean of 200 and a standard deviation of 10, describing how the results are expected to distribute around the mean through multiple experiments.

Normal Distribution

The normal distribution, often referred to as the bell curve due to its shape, is a probability distribution that is commonly used in statistics. It's defined by its mean (center point) and standard deviation (spread or width of the curve).

Thanks to the central limit theorem, the normal distribution can be used to approximate the behavior of various data types. This theorem states that when a large sample size is collected, the sample mean will approximately follow a normal distribution, even if the original data isn't normally distributed.

• In our exercise, we apply the central limit theorem to the binomial distribution from the coin tosses.
• The number of heads becomes a random variable that demonstrates a normal pattern due to the large number of tosses, which is 400.

We thus treat our results as fitting into a normal distribution with a mean \( \mu = 200 \) and standard deviation \( \sigma = 10 \). Once this transformation is made, we gain access to various mathematical tools like the standard normal table to find probabilities easily.

Probability Calculation

Calculating probabilities using the normal distribution is a standard technique in statistics. It involves transforming our actual variable into a standardized score, called a \( z \)-score, that can be compared against a known standard distribution.

• To find the probability of obtaining at most 190 heads, we first convert that number into a standardized \( z \)-score: \( z = \frac{190 - 200}{10} = -1.0 \).
• This score tells us how many standard deviations away 190 is from the mean of 200.

With this \( z \)-score, we can lookup the probability in a standard normal distribution table. Initially, we found a probability of 0.1587 without the histogram correction.

However, to achieve a more precise approximation, we include the continuity correction. This shifts our boundary slightly by considering 190.5 instead of 190, leading to a \( z \)-score of \( -0.95 \) and a corrected probability of approximately 0.1711. This small adjustment improves the approximation in cases where a cumulative property (like a count of coin heads) is approximated by a continuous function.