Survey

Problem

In a city the election of the mayor is approaching. 12% of the eligible voters are young voters, i.e. persons aged between 18 and 24 years. Before the election campaign, a representative poll amongst the eligible voters is conducted. According to the poll, 44% of the polled eligible voters have already decided in favour of a candidate. One out of seven of the respondents, who have not yet decided in favour of a candidate, is a young voter.

The following outcomes are considered:

\(J\): “A person randomly selected from the respondents is a young voter.”

\(K\): “A person randomly selected from the respondents has decided already in favour of a candidate.”

1. Compile a completely filled fourfold table for the context described above.

2. Demonstrate that \(P_J(\overline{K})>P_{\overline{J}}(\overline{K})\). Justify that, in spite of validity of this inequality, it is not reasonable to concentrate predominantly on young voters in the election campaign.

3. On a specific day during his campaign, the candidate of party A speaks to 48 randomly selected eligible voters. Determine the probability that there are exactly six young voters amongst them.

Solution of part a

The problem yields the following fourfold table:

	\(K\)	\(\overline{K}\)	\(\sum\)
\(J\)		\(X\)	12%
\(\overline{J}\)		\(6\cdot X\)	88%
\(\sum\)	44%	56%	100%

The statement “One out of seven respondents, who have not yet decided in favour of a candidate, is a young voter.” is considered in the column \(\overline{K}\). The sum over the yet undecided eligible voters yields \(X=8\%\). The empty entries in the rows \(J\) and \(\overline{J}\) can be completed by subtraction.

	\(K\)	\(\overline{K}\)	\(\sum\)
\(J\)	4%	8%	12%
\(\overline{J}\)	40%	48%	88%
\(\sum\)	44%	56%	100%

Solution of part b

To show the inequality, we compute \(P_J(\overline{K})\) and \(P_{\overline{J}}(\overline{K})\).

\[\begin{split}P_J(\overline{K}) = \frac{P(\overline{K} \cap J)}{P(J)} = \frac{8\%}{12\%} = 66.7\% \\ P_{\overline{J}}(\overline{K}) = \frac{P(\overline{K} \cap \overline{J})}{P(\overline{J})} = \frac{48\%}{88\%} \approx 54.5\% \\\end{split}\]

The inequality \(P_J(\overline{K})>P_{\overline{J}}(\overline{K})\) is thus fulfilled. Nevertheless, it is not reasonable to concentrate on the young voters during the election campaign. For the outcome of the election, the total number of voters has to be considered. Even though by percentage, more older voters than young voters have already decided in favour of a candidate, this is not the case in absolute numbers. At 8% young voters and 48% older voters, who have not yet taken their decision, the election campaign should better focus on older voters.

Solution of part c

The probability for an eligible voter to be a young voter is 12%. The probability that there are exactly six young voters amongst 48 voters can be determined from the binomial distribution:

\[P^{48}_{0.12}(6) = {48 \choose 6} \cdot 0.12^6 \cdot (1-0.12)^{42} = 17.07\%\]

We can simulate this experiment with Sage.