Problem 26 - A First Course in Probability

Question

Two types of coins are produced at a factory: a fair coin and a biased one that comes up heads 55 percent of the time. We have one of these coins but do not know whether it is a fair coin or a biased one. In order to ascertain which type of coin we have, we shall perform the following statistical test: We shall toss the coin 1000 times. If the coin lands on heads 525 or more times, then we shall conclude that it is a biased coin, whereas if it lands on heads fewer than 525 times. then we shall conclude that it is a fair coin. If the coin is actually fair, what is the probability that we shall reach a false conclusion? What would it be if the coin were biased?

Step-by-Step Solution

Answer

For the fair coin, the probability of reaching a false conclusion is: \[ P(\text{False Conclusion}) \approx 0.057 \] For the biased coin, the probability of reaching a false conclusion is: \[ P(\text{False Conclusion}) \approx 0.021 \]

Step 1: Identify the given information

We are given the following information: - \(n = 1000\) (number of tosses) - For a fair coin, the probability of success (\(p\)) is 0.5. - For a biased coin, the probability of success (\(p\)) is 0.55. - The threshold for determining the type of coin is 525.

Step 2: Calculate the probability for the fair coin

We are asked to find the probability of the fair coin leading to 525 or more heads. To find this probability, we need to sum the probabilities of obtaining exactly 525 heads, 526 heads, and so on, up to 1000 heads. So, we can represent this probability as: \[ P(X \geq 525) = \sum_{k=525}^{1000} P(k) \] where \(P(k)\) is computed using the binomial probability formula mentioned earlier, and \(p = 0.5\) for the fair coin. Since calculating this sum manually would be very time-consuming, we can use calculator tools or statistical software to compute this probability.

Step 3: Calculate the probability for the biased coin

Similar to the fair coin, we need to find the probability of the biased coin leading to 525 or more heads. So, we can represent this probability as: \[ P(X \geq 525) = \sum_{k=525}^{1000} P(k) \] where \(P(k)\) is computed using the binomial probability formula mentioned earlier, and \(p = 0.55\) for the biased coin. Again, we can use calculator tools or statistical software to compute this probability.

Step 4: Calculate the probability of reaching a false conclusion

To calculate the probability of reaching a false conclusion for both coins, we need to compare our calculated probabilities to the threshold of 525. For the fair coin, the probability of reaching a false conclusion is: \[ P(\text{False Conclusion}) = P(X \geq 525) \] For the biased coin, the probability of reaching a false conclusion is: \[ P(\text{False Conclusion}) = P(X < 525) \] Note that for the biased coin, the probability of reaching a false conclusion is essentially the complement of the probability obtained in step 3, since we are looking for the probability of fewer heads than 525. Now, you can use the computed probabilities to determine the probability of reaching a false conclusion for both coins.

Key Concepts

Binomial Distribution

The concept of a binomial distribution is central to understanding how we analyze coin tosses in statistics. When we say a distribution is binomial, we mean that it arises from situations where there are two possible outcomes, such as heads or tails in a coin toss.

Key features of a binomial distribution include:

• It depends on a fixed number of trials, denoted as \(n\). In our case, this is 1000 coin tosses.
• Each trial is independent, meaning the outcome of one toss doesn't affect another.
• The probability of success, \(p\), remains constant for each trial. For a fair coin, \(p = 0.5\), and for the biased coin, \(p = 0.55\).

To calculate the probability of a certain number of heads, we use the binomial probability formula, which requires computational tools for large calculations. This is because finding the sum of probabilities for 525 to 1000 heads can be complex without software like a statistical calculator or programming languages such as Python.

Probability of False Conclusion

In hypothesis testing, one essential consideration is the probability of reaching a false conclusion. A false conclusion happens when we make the wrong decision based on our test results.

For our exercise:

• With the fair coin, a false conclusion occurs if we think it is biased, meaning we find 525 or more heads.
• For the biased coin, a false conclusion means assuming it is fair, implying we find less than 525 heads.

These errors help us understand Type I and Type II errors in statistical tests:

• Type I Error: Rejecting a true null hypothesis, like calling a fair coin biased.
• Type II Error: Failing to reject a false null hypothesis, like deeming a biased coin fair.

Calculating these probabilities helps us appreciate the limitations of the test used and the likelihood of making incorrect calls.

Fair vs Biased Coin

Understanding the difference between a fair and a biased coin is vital in this problem. A fair coin is one where heads and tails each have a 50% chance of occurring, making these outcomes equally likely.

Characteristics:

• Fair Coin: \(p = 0.5\) for both heads and tails.
• Biased Coin: Here, \(p = 0.55\) for heads, suggesting a slight preference for landing heads.

In our test, determining the type of coin after 1000 tosses relies on comparing the results to a threshold of 525 heads. This threshold helps distinguish between fair and biased coins based on statistical likelihood.

Even a small bias like 5% can significantly affect outcomes, showcasing how subtle differences impact probability and decision making.

Statistical Decision Making

Statistical decision making involves choosing between hypotheses based on probability calculations and observed data. Here, we use statistical tests to decide if a coin is fair or biased based on 1000 tosses.

The decision rules are:

• Conclude the coin is biased if there are 525 or more heads.
• Conclude the coin is fair otherwise.

This process involves critical thinking and statistical knowledge to set appropriate thresholds and interpret outcomes. In practical applications, this could influence product quality control or experimental results interpretation.

Ultimately, statistical decision making is about balancing risks, benefits, and likelihoods to make informed choices, proving valuable in various fields like research, business, and science.