Aufgaben:Exercise 2.1: Election Demand

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Result of election demand

In an election, the three candidates  $A$,  $B$  and  $C$  are running for mayor.

• The candidate who receives more than  $50\%$  of the votes cast is elected.
• If none of the three candidates succeeds in the first ballot, a runoff election shall be held between the two candidates with the most votes.

Immediately after the closing of the polling stations, the result of an election demand shall be presented:

Candidate  $A$:   $48\%$,       Candidate  $B$:   $30\%$,       Candidate  $C$:   $22\%$.

This demand is based on a survey of only  $N = 2000$  of the total  $N' = 800 \hspace{0.05cm}000$  voters.

In answering the following questions, assume the following:

• The actual (percentage) votes obtained in the election by candidates  $A$,  $B$  and  $C$  can be taken as the probabilities  $p_{\rm A}$,  $p_{\rm B}$  and  $p_{\rm C}$  although these are also themselves determined as relative frequencies  $($related to  $N')$  .
• The  $2000$  selected voters ideally represent the entire electorate in a statistical sense and answered truthfully when asked to vote.
• According to Bernoulli's Law of Large Numbers  the results of this demand are to be understood as relative frequencies:

$$h_{\rm A} = 0.48,\hspace{0.8cm}h_{\rm B} = 0.30,\hspace{0.9cm} h_{\rm C} = 0.22.$$

Hints:

• The exercise belongs to the chapter  From Random Experiment to Random Variable.

• The topic of this chapter is illustrated with examples in the  (German language)  learning video  Bernoullisches Gesetz der großen Zahlen  $\Rightarrow$   Bernoulli's Law of Large Numbers.

Questions

1

Who do you expect as future mayor after this demand?

	candidate  $A$,
	Candidate  $B$,
	Candidate  $C$.

2

What is the probability that no runoff will be required?  Specify the upper bound here.

$\text{Maximum: Pr(no runoff)} \ = \ $ $\ \rm \%$

3

We now assume that candidate  $A$  actually receives exactly  $48\%$  of the votes.
What is the probability (at most) that candidate $C$ will reach the runoff?

$\text{Maximum: Pr(}C \ \text{in runoff)}\ = \ $ $\ \rm \%$

Musterlösung

Solution

(1)  One should at least believe this demand that  $\underline{\text{candidate} \ A}$  is likely to win.

(2)  The probability that the demand  $(h_{\rm A})$  differs from the final outcome  $(p_{\rm A})$  by more than  $2\%$  is, according to Bernoulli's law of large numbers, with  $N = 2000$: $${\rm Pr}(|h_{\rm A} - p_{\rm A}| \geq 0.02) \leq \frac{1}{4 \cdot 2000\cdot 0.02^2} = 0.3125.$$

• This probability includes the two equally likely cases that  $p_{\rm A} \le 46\%$  and  $p_{\rm A} \le 50\%$  is.
• Only in the last case there is no runoff:

$${\rm Pr(no\hspace{0.1cm}runoff)} \le 0.156 \hspace{0.15cm}\underline{=15.6 \%}.$$

(3)  With  $\varepsilon = 4\%$  $($is given by  $0.26 -0.22)$  yields the law of large numbers:

$${\rm Pr}\left(|h_{\rm C}-p_{\rm C}|\ge 0.04\right)\le\rm\frac{1}{4\cdot 2000\cdot 0.04^2}=0.078.$$

It follows that:

• The probability that candidate  $C$  receives at least  $26\%$  of the votes is not greater than  $3.9\%$.
• Since  $p_{\rm A} = 0.48$  was assumed fixed, in this case simultaneously  $p_{\rm B} \le 0.26$.
• Since these are continuous random variables,  $(p_{\rm C} \ge 0.26, \; p_{\rm B} \le 0.26)$  and  $(p_{\rm C} > 0.26, \; p_{\rm B}< 0.26)$  are equal.
• Thus, the probability that  $C$  reaches the runoff is also limited to  $3.9\%$  :

$${\rm Pr(}C\rm \hspace{0.1cm}reaches\hspace{0.1cm}runoff \hspace{0.1cm}election)\le 0.039 \hspace{0.15cm}\underline{= 3.9 \%}.$$