Problem 121 - College Algebra Essentials

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

A tossed coin is considered unfair if the number of times it results in heads when tossed 100 times is 59 or more, or 41 or less.

Step 1: Interpret the Given Inequality

The given inequality is presented in absolute value form, which means the result could be either positive or negative. As such, it defines a range of values.

Step 2: Solve the Inequality for Positive Case

When the inside of the absolute value is positive, the given inequality becomes \( \frac{h-50}{5} \geq 1.645 \). Multiply each side by 5 to isolate \( h \), resulting in \( h \geq 50 + 5*1.645 = 58.225 \)

Step 3: Solve the Inequality for Negative Case

When the inside of the absolute value is negative, the given inequality becomes \( - \frac{h-50}{5} \geq 1.645 \). Once again, multiply each side by 5 and solve for \( h \), resulting in \( h \leq 50 - 5*1.645 = 41.775 \)

Step 4: Conclude the Results

Since a coin cannot be tossed a fraction of a time, round off to the nearest whole number. Therefore, a coin is considered unfair if it is tossed 100 times and the outcomes that result in heads are '59 and more' or '41 and less'.