Question

complete the tables for different values of \hat{p} and \hat{q}=1 - \hat{p}. from the tables, which value of \hat{p} appears to give the maximum value of the product \hat{p}\hat{q}.
complete the table below.
\

$$\begin{tabular}{|c|c|c|} \\hline \\hat{p} & \\hat{q}=1 - \\hat{p} & \\hat{p}\\hat{q} \\ \\hline 0.0 & 1.0 & 0.00 \\ \\hline 0.1 & 0.9 & 0.09 \\ \\hline 0.2 & 0.8 & \\square \\ \\hline 0.3 & \\square & \\square \\ \\hline 0.4 & \\square & \\square \\ \\hline 0.5 & \\square & \\square \\ \\hline \\end{tabular}$$

(type integers or decimals.)

Explanation:

Step1: Calculate \(\hat{q}\) for \(\hat{p} = 0.3\)

Given \(\hat{q}=1 - \hat{p}\), when \(\hat{p}=0.3\), then \(\hat{q}=1 - 0.3=0.7\)

Step2: Calculate \(\hat{p}\hat{q}\) for \(\hat{p} = 0.3\)

\(\hat{p}\hat{q}=0.3\times0.7 = 0.21\)

Step3: Calculate \(\hat{q}\) for \(\hat{p} = 0.4\)

When \(\hat{p}=0.4\), \(\hat{q}=1 - 0.4 = 0.6\)

Step4: Calculate \(\hat{p}\hat{q}\) for \(\hat{p} = 0.4\)

\(\hat{p}\hat{q}=0.4\times0.6=0.24\)

Step5: Calculate \(\hat{q}\) for \(\hat{p} = 0.5\)

When \(\hat{p}=0.5\), \(\hat{q}=1 - 0.5=0.5\)

Step6: Calculate \(\hat{p}\hat{q}\) for \(\hat{p} = 0.5\)

\(\hat{p}\hat{q}=0.5\times0.5 = 0.25\)

Answer:

\(\hat{p}\)	\(\hat{q}=1-\hat{p}\)	\(\hat{p}\hat{q}\)
\(0.1\)	\(0.9\)	\(0.09\)
\(0.2\)	\(0.8\)	\(0.16\)
\(0.3\)	\(0.7\)	\(0.21\)
\(0.4\)	\(0.6\)	\(0.24\)
\(0.5\)	\(0.5\)	\(0.25\)

The value of \(\hat{p}\) that gives the maximum value of the product \(\hat{p}\hat{q}\) is \(\hat{p}=0.5\)