Question

A coin is spun 25 times. Let \(x\) be the number of spins that result in heads (H). Consider the following rule for deciding whether or not the coin is fair: Judge the coin fair if \(8 \leq x \leq 17\). Judge the coin biased if either \(x \leq 7\) or \(x \geq 18\). a. What is the probability of judging the coin biased when it is actually fair? b. What is the probability of judging the coin fair when \(P(\mathrm{H})=.9\), so that there is a substantial bias? Repeat for \(P(\mathrm{H})=.1 .\) c. What is the probability of judging the coin fair when \(P(\mathrm{H})=.6\) ? when \(P(\mathrm{H})=.4 ?\) Why are the probabilities so large compared to the probabilities in Part (b)? d. What happens to the "error probabilities" of Parts (a) and (b) if the decision rule is changed so that the coin is judged fair if \(7 \leq x \leq 18\) and unfair otherwise? Is this a better rule than the one first proposed? Explain.

Short answer

The calculations primarily involve calculating probabilities using the binomial distribution formula. The final answer depends on the interpretations of these calculated probabilities under different conditions.

Step-by-step solution

Understanding the problem

Define \(X\) as a random variable denoting the number of heads obtained from 25 coin flips. A fair coin implies \(P(H) = 0.5\). A decision rule has also been given to determine if the coin is fair or biased. If \(8 \leq X \leq 17\), the coin is judged to be fair; Otherwise, it's judged to be biased.

a) Probability of judging a fair coin biased

Calculate the probability that the coin is judged biased when it's actually fair. This translates to finding \(P(X \leq 7) + P(X \geq 18)\) when the coin is fair. It implies we compute the sum of probabilities of getting less than 8 and more than 17 heads. Use the binomial probability formula: \(P(X=k) = C(n,k)*(p^k)*((1-p)^(n-k))\) where \(n\) is the number of trials, \(k\) is the number of successes, \(p\) is the probability of success (here assumed to be 0.5 for a fair coin). Compute these probabilities.

b) Probability of judging a biased coin fair

The probability of judging the coin fair will depend on the coin's actual bias. Here calculate \(P(8 \leq X \leq 17)\) when \(P(H) = 0.9\) and \(P(H) = 0.1\). It implies we compute the cumulative probabilities of getting between 8 and 17 heads (both inclusive), when the probability of getting heads is respectively 0.9 and 0.1. Again, use the binomial probability formula and compute these probabilities.

c) Probability of judging a biased coin fair

Similarly, find the probability \(P(8 \leq X \leq 17)\) when \(P(H) = 0.6\) and \(P(H) = 0.4\). These numbers are closer to 0.5, hence the probabilities will be larger as compared to when the coin is heavily biased.

d) Effect of changing the rule on error probabilities

In this step, investigate how changing the decision rule to \(7 \leq X \leq 18\) affects the error probabilities computed earlier. To comment on the quality of this new rule, you will need to compare the error probabilities from this rule with the ones from the original rule. If the error probabilities decrease, the new rule would be judged as a better one, subject to other considerations.

Key concepts

Binomial Probability

Understanding binomial probability is crucial when dealing with exercises like the fair coin problem. In its essence, a binomial probability represents the likelihood of a specific number of successes in a fixed number of independent trials, each with the same probability of success.

Let’s consider an experiment, such as flipping a coin, where there are only two possible outcomes – heads or tails. If the coin is fair, each outcome has an equal probability of occurring, which is 0.5 or 50%. Binomial probability comes into play when we want to know, for instance, the probability of getting exactly 4 heads in 10 flips. For a scenario with a fair coin, this can be calculated using the binomial probability formula:
\[\begin{equation}P(X=k) = \binom{n}{k} \times p^k \times (1-p)^{n-k}\right\end{equation}\]
where:

• \(n\) is the number of trials (coin flips)
• \(k\) is the number of successful trials (getting heads)
• \(p\) is the probability of success on a single trial (probability of getting heads)

In a real-life scenario, applying binomial probability can help determine whether a coin is biased by comparing observed outcomes against expected binomial distributions.

Random Variable

In probability and statistics, a random variable is fundamental for quantifying outcomes of stochastic processes, such as flipping a coin, rolling a die, or drawing a card. A random variable is essentially a function that assigns a numerical value to each possible outcome of a random process.

In our exercise, the random variable \(X\) denotes the number of heads resulting from flipping a coin 25 times. Here, \(X\) can take any value between 0 and 25, inclusive. When a random variable is defined, we can discuss its probability distribution, which is a listing of all the possible values that the random variable can take and the probability associated with each of these values. In the case of flipping a fair coin, the probability distribution of \(X\) is symmetric, because the probability of getting a head is equal to the probability of getting a tail on each flip. However, if the coin were biased, the distribution would skew towards more heads or tails, depending on the bias.

Fair Coin Bias

When discussing probability, the concept of a 'fair coin' is frequently used as a benchmark. A fair coin means that there are no biases, and hence, the probability of landing heads is equal to that of tails, both at 0.5. If a coin is biased, the probabilities differ from 0.5.

The fair coin assumption allows us to model situations and calculate probabilities based on a 50-50 chance. However, in the provided exercise, we're challenged to determine if a coin is fair based on the number of heads yielded in 25 flips. If the coin is genuinely fair, the number of heads should, most of the time, fall within a certain range, which is determined by the binomial distribution.

Considering Bias in Decision Rules

When a decision rule is established, like in our exercise, judging a coin fair if \(8 \right\leq X \right\leq 17\), we are using probabilities to make inferences about the coin's fairness. But such rules are not perfect; they come with error probabilities, which are the chances of incorrectly identifying a fair coin as biased and vice versa. Fine-tuning these decision rules involves balancing sensitivity with specificity—distinguishing fair coins from biased ones without excessive errors.