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Chapter 1: Problem 121

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.

Short Answer

Expert verified
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 by step solution

01

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.
02

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 \( h \geq 50 + 5*1.645 = 58.225 \)
03

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 \( h \leq 50 - 5*1.645 = 41.775 \)
04

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'.

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