Problem 99 - College Algebra

Question

If a coin is tossed 100 times, we would expect approximately 50 of the outcomes to be heads. It can be demonstrated that a coin is unfair if \(h,\) the number of outcomes that result in heads, satisfies \(\left|\frac{h-50}{5}\right| \geq 1.645\) Describe the number of outcomes that determine an unfair coin that is tossed 100 times.

Step-by-Step Solution

Answer

The number of outcomes that would determine an unfair coin is all the whole numbers from 59 to 100 inclusive, and all the whole numbers from 41 down to 0 inclusive.

Step 1: Solve the Inequality for h

The given inequality can be decomposed into two separate inequalities due to the modulus sign i.e., \(\frac{h-50}{5} \geq 1.645\) and \(\frac{h-50}{5} \leq -1.645\). Now, each inequality can be solved separately to find 'h'. Based on the mathematical rules, for the first inequality, multiply through by 5 to remove the denominator and then add 50 to both sides. Similarly, for the second inequality.

Step 2: Find the Range for h

From the first inequality, h is greater than or equal to \(1.645*5 + 50 = 58.225\). From the second, h is less than or equal to \(-1.645*5 + 50 = 41.775\). As 'h' should be a whole number (since it represents the number of outcomes), we have to round these. Hence, if the number of outcomes that result in heads is 59 or more, or 41 or less, then the coin can be confirmed to be unfair.

Step 3: Conclusion of the Problem

So the range of outcomes that would determine an unfair coin is all the whole numbers starting from 59 till 100 and from 41 down to 0.

Key Concepts

Inequalities

Inequalities are an important part of mathematics and statistics. They help us determine ranges of values that satisfy certain conditions. In this exercise, an inequality is used to detect if a coin is unfair. The inequality given is \(\left|\frac{h-50}{5}\right| \geq 1.645\).
It involves a calculation that compares the difference between the observed number of heads \(h\) and the expected number 50, assuming a fair coin. The absolute value ensures that we are only interested in the magnitude of the difference, not its direction.
To solve this, we split it into two separate inequalities: \(\frac{h-50}{5} \geq 1.645\) and \(\frac{h-50}{5} \leq -1.645\). Each of these represents a potential condition for the coin being unfair, and solving them gives us the critical values for \(h\). This approach is very practical in statistics, especially to verify hypotheses or to identify biases.

Statistics

Statistics is the science of collecting, analyzing, and interpreting data. It helps us draw conclusions about real-world phenomena. In this problem, we rely on statistical principles to decide if a coin is fair or unfair. When we toss a fair coin numerous times, ideally, half the tosses should result in heads.
In the exercise, 100 coin tosses are conducted to see how they compare with the expected outcomes. The statistical concept behind this is the idea of a normal distribution. For a fair 100 tosses, we expect 50 heads, but due to randomness, the actual results might vary slightly.
The inequality \(\left|\frac{h-50}{5}\right| \geq 1.645\) is linked with a concept from statistics known as a standard normal distribution. It’s a statistical method that helps determine the probability that an observed number of heads \(h\) falls within a certain range around the expected value. If the number falls outside of this range, it suggests some deviation from fairness.

Coin flipping

Coin flipping is a simple probability exercise where the outcomes should ideally be 50% heads and 50% tails — a basic example of a binary event. Each flip of the coin is independent and identically distributed, which means that the result of one flip doesn't affect another.
In terms of probability, each side has an equal chance of landing face up, thus when we flip a coin 100 times, we expect to get about 50 heads. However, in practice, we might get a slightly different number of heads, and this is where probability helps us understand what deviations might be considered typical.
The exercise at hand uses coin flipping as a way to explore whether a coin might be fair or biased. By setting up the inequality \(\left|\frac{h-50}{5}\right| \geq 1.645\), we assess various outcomes. If 59 or more, or 41 or fewer heads come up, then it indicates that the coin is likely unfair. This specific test can be linked to statistical tests like the z-test, which helps determine if a sample deviates significantly from expectations.