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 less 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

\(P(False \hspace{1mm} Conclusion \hspace{1mm} | \hspace{1mm} Fair \hspace{1mm} Coin) ≈ 0.0568\) #Step 3: Calculate the Probability of False Conclusion if the Coin is Biased# #tag_title#Determine the Range for False Conclusion#tag_content#To find the range to compute the probability, we must first determine when we will falsely conclude that the coin is fair. With a biased coin, we are looking for a result with less than 525 heads in 1000 tosses. We need to calculate the cumulative probability for landing heads for 0 to 524 times. #tag_title#Calculate the Cumulative Probability of 0-524 Heads#tag_content#We calculate the cumulative probability using the binomial probability formula: \(P(X=k) = C(n, k) \cdot p_B^k \cdot (1-p_B)^{n-k}\) Compute \(P(X\leq 524)\) with n = 1000, and \(p_B = 0.55\). \(P(False \hspace{1mm} Conclusion \hspace{1mm} | \hspace{1mm} Biased \hspace{1mm} Coin) = \sum_{k=0}^{524} C(1000, k) \cdot p_B^k \cdot (1-p_B)^{1000-k}\) Using a computing tool (such as calculator or Python), we can calculate the probability that the test will mistakenly declare the biased coin as fair. \(P(False \hspace{1mm} Conclusion \hspace{1mm} | \hspace{1mm} Biased \hspace{1mm} Coin) ≈ 0.1139\) In conclusion, if the coin is actually fair, the probability of reaching a false conclusion is approximately 5.68%. If the coin is biased, the probability of reaching a false conclusion is approximately 11.39%.

Step 1: Determine the Range for False Conclusion

To find the range to compute the probability, we must first determine when we will falsely conclude that the coin is biased. With a fair coin, we are looking for a result with 525 or more heads in 1000 tosses. We need to calculate the cumulative probability for landing heads for 525 to 1000 times.

Step 2: Calculate the Cumulative Probability of 525-1000 Heads

We calculate the cumulative probability using the binomial probability formula: \(P(X=k) = C(n, k) \cdot p_F^k \cdot (1-p_F)^{n-k}\) Compute \(P(X\geq 525)\) with n = 1000, and \(p_F = 0.5\). We can use the complement rule to find the probability of at least 525 heads: \(P(X\geq 525) = 1 - P(X\leq 524)\) Then, we need to calculate the cumulative probability for all k values ranging from 0 to 524: \(P(False \hspace{1mm} Conclusion \hspace{1mm} | \hspace{1mm} Fair \hspace{1mm} Coin) = \sum_{k=0}^{524} C(1000, k) \cdot p_F^k \cdot (1-p_F)^{1000-k}\) Using a computing tool (such as calculator or Python), we can calculate the probability that the test will mistakenly declare the fair coin as biased.

Key Concepts

Statistical Tests

Statistical tests are methods used to make decisions based on data. These decisions are often about whether to reject or accept a hypothesis. In the context of the coin-tossing experiment, we are using a statistical test to determine if a coin is fair or biased.
Here's how it works:

• We have two hypotheses — the null hypothesis (the coin is fair) and the alternative hypothesis (the coin is biased).
• The test involves tossing the coin 1000 times and observing the results.
• If the number of heads is 525 or more, we reject the null hypothesis and conclude that the coin is biased.
• If it's less than 525, we accept the null hypothesis and conclude that the coin is fair.

The aim is to use the gathered data to make a reasonable decision about the type of coin we have.

Probability of Error

Whenever we perform a statistical test, there's a chance of making an error. This is because we're making conclusions based on sample data rather than the entire population.
In our coin experiment:

• A Type I error happens if we incorrectly conclude the coin is biased when it's actually fair.
• A Type II error would occur if we concluded the coin is fair when it's actually biased.

The task involves finding the probability of making such an error. For the fair coin, this means determining the likelihood of getting 525 or more heads (a false conclusion). This is calculated using cumulative probability, giving us an insight into how often our test might lead us astray.

Cumulative Probability

Cumulative probability is a concept that helps quantify the likelihood of a range of outcomes occurring. In this scenario, it helps us understand the aggregate probability of observing 525 or more heads if the coin is fair.
By using cumulative probability:

• We add up the probabilities of all outcomes from 525 heads to 1000 heads.
• To simplify, we use the complement rule: calculate the probability for 0 to 524 heads, then subtract it from 1.

This calculation provides the aggregate probability of obtaining a false positive result — concluding the coin is biased, when it is fair. Calculating cumulative probabilities often requires computational tools due to the complexity of combining many individual probabilities.

Binomial Probability Formula

The binomial probability formula is crucial for calculating the probability of obtaining a given number of heads in coin tosses. It is defined as:\[P(X=k) = C(n, k) \cdot p^k \cdot (1-p)^{n-k}\]- Here, \(C(n, k)\) is the combination of \(n\) items taken \(k\) at a time, which calculates the number of ways to have \(k\) heads in \(n\) tosses.- \(p\) is the probability of success (landing heads), and \(n\) and \(k\) are the number of trials and successes, respectively.For a fair coin where \(p = 0.5\), this formula allows us to calculate the probabilities necessary for determining whether the coin is fair or not. Utilizing this formula repeatedly, and then summing the results for a range of successes, helps us find cumulative probabilities efficiently.