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.

Key Concepts

Probability Distribution

A probability distribution gives us a complete picture of the possible outcomes of a random experiment and how likely each outcome is. For discrete events, like tossing a coin, this distribution lists all possible outcomes and the probability of each happening. In the context of this exercise, we are dealing with the binomial distribution, a specific type of probability distribution.

The binomial distribution is used when an experiment meets specific criteria: there are a fixed number of trials, each trial is independent, there are only two possible outcomes (success or failure), and the probability of success is constant across trials. In our coin toss example:

• Number of trials (n) = 10,000
• Two outcomes: Heads (success) or Tails (failure)
• Probability of success (p) = 0.5, if we assume the coin is fair.

Understanding probability distributions ensures we can predict and analyze real-world events, helping us make informed decisions based on observed data.

Bernoulli Trials

Bernoulli trials are the building blocks of binomial distributions. Each trial in a Bernoulli process can only lead to a success or failure outcome. These trials must have a constant probability of success across all trials and are independent of each other.

When tossing a fair coin, each coin flip is a Bernoulli trial. Here's why:

• Each flip is independent: the outcome of one flip does not affect the next.
• There are two possible outcomes: heads or tails.
• The probability of success (heads) is constant: 0.5 for each toss.

When we put together many Bernoulli trials, such as the 10,000 coin flips in our example, we see the characteristic bell curve of the binomial distribution forming. This is because the independent trials sum up to create a composite probability distribution, where we can use binomial formulas to calculate outcomes, like the likelihood of getting a certain number of heads.
Understanding Bernoulli trials helps us appreciate how randomness operates on a small scale, setting the foundation for more complex probability problems.

Standard Deviation

The standard deviation is an important measure in statistics that tells us how spread out the values in a data set are. In terms of probability distributions, it indicates how much the number of successes in Bernoulli trials varies from the mean.

For a binomial distribution, the formula for standard deviation (\(\sigma\)) is:\(\sigma = \sqrt{n \cdot p \cdot (1-p)}\), where n is the number of trials and p is the probability of success. From our exercise:

• n = 10,000
• p = 0.5

Substituting these values, we get:\(\sigma = \sqrt{10000 \cdot 0.5 \cdot 0.5} = \sqrt{2500} = 50\).

• This means in a fair scenario, we expect most outcomes to lie within 50 heads of the mean (5000 heads).

Standard deviation helps determine the range of normal variation in results, setting the criterion for outcomes we perceive as unusual or significant.

Mean in Probability

In probability, the mean or expected value represents the "center" of a probability distribution. It is the average number of successes over a large number of trials. For a binomial distribution, the mean (\(\mu\)) is given by the formula: \(\mu = n \cdot p\).

In the coin tossing exercise, calculating the mean gives us:

• n = 10,000
• p = 0.5

Applying these values:\(\mu = 10000 \cdot 0.5 = 5000\).

• This means we would expect to get heads 5,000 times with a fair coin.

Understanding the mean is crucial because it provides a point of reference, showing where the center of your data should be if conditions (like a fair coin) hold true. When outcomes in an experiment deviate significantly from the mean, it could suggest an underlying irregularity or bias, as suspected in the exercise where 5,800 heads were observed—a number far outside the expected range given our calculated standard deviation.