Problem 27 - A First Course in Probability

Question

In 10,000 independent tosses of a coin, the coin landed on heads 5800 times. Is it reasonable to assume that the coin is not fair? Explain.

Step-by-Step Solution

Answer

The observed number of heads (5800) is outside the acceptable range (4900 to 5100), which is based on two standard deviations from the mean (\(5000 \pm 100\)) for a fair coin. Therefore, it is reasonable to assume that the coin is not fair.

Step 1: Definition of Binomial Distribution

A binomial distribution refers to the probability distribution of a discrete random variable which describes the number of successes in a fixed number of independent Bernoulli trials with the same probability of success. In this exercise, we have 10,000 independent tosses (n) of a coin, and if the coin were fair, the probability of landing heads would be 0.5 (p).

Step 2: Determine the mean and standard deviation of a binomial distribution

Calculating the mean (\(\mu\)) and standard deviation (\(\sigma\)) of a binomial distribution is essential to determine if the observed number of heads is within the acceptable range or not. The formula for the mean of a binomial distribution is given by: \(\mu = n*p\) The formula for the standard deviation is given by: \(\sigma = \sqrt{n*p*(1-p)}\) In the current problem, we have \(n = 10000\) and \(p = 0.5 \).

Step 3: Calculate the mean and standard deviation

Using the formulas from Step 2, we can calculate the mean and standard deviation for our problem: Mean: \(\mu = 10000 * 0.5 = 5000\) Standard deviation: \(\sigma = \sqrt{10000 * 0.5 * (1-0.5)} = \sqrt{10000 * 0.5 * 0.5} = \sqrt{2500}\) = 50

Step 4: Apply the rule of thumb

A common rule of thumb for determining whether an observed outcome is within an acceptable range of the mean is by checking if it falls within two standard deviations from the mean. Using this rule of thumb, the acceptable range for the number of heads is given by: Lower limit: \(\mu - 2\sigma = 5000 - 2 * 50 = 4900\) Upper limit: \(\mu + 2\sigma = 5000 + 2 * 50 = 5100\)

Step 5: Compare the observed number of heads with the acceptable range

Now we have the acceptable range (4900 to 5100) for the number of heads, we can compare the observed number of heads (5800) with this range. The observed number of heads (5800) is outside the acceptable range (4900 to 5100). Therefore, it is reasonable to assume that the coin is not fair.