91Ó°ÊÓ

Referring to example 7.3.2, it must be true that \(V \le 0.01\) after 22 items have been selected.

Let,

\(\gamma = \left( {y + 1} \right)\left( {y + z + 2} \right)\)

Then,

\(1 - \gamma = \left( {z + 1} \right)\left( {y + z + 2} \right)\,\,{\rm{and}}\,\,V = \frac{{\gamma \left( {1 - \gamma } \right)}}{{\left( {y + z + 3} \right)}}\)

The maximum value of\(\gamma \left( {1 - \gamma } \right)\)is\(\frac{1}{4}\)and is attained when\(\gamma = \frac{1}{2}\)

Therefore,

\(\begin{aligned}V \le \frac{{\frac{1}{4}}}{{\left( {y + z + 3} \right)}}\\ \le \frac{1}{{\left( {4\left( {y + z + 3} \right)} \right)}}\end{aligned}\)

It now follows that if\(\frac{1}{{\left( {4\left( {y + z + 3} \right)} \right)}} \le 0.01\,\,\,then\,\,\,V \le 0.01\)

But the first inequality will be satisfied if\(y + z \ge 22\)

Since,\(y + z\)is the total number of items that have been selected, it follows that this number need not exceed 22.

Hence proved.