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Reality and statistics
Example 1
When evaluating the result of the elections after counting the \latex{ 90 }% of the votes the distribution of the votes given for the \latex{ 3 } candidates (\latex{ A }, \latex{ B }, \latex{ C }) is shown in the pie chart. (Figure 14). (The candidate who gets the most votes wins the elections.)
- Can we say after this that candidate \latex{ C } will surely win?
- Can candidate \latex{ A } win the elections?
- What percentage of the lead shall candidate \latex{ C } have compared to the candidate behind him so that he can be sure of his winning even mathematically with \latex{ 90 }% of the votes processed?
invalid
\latex{ C }
\latex{ B }
\latex{ A }
\latex{ 26 }%
\latex{ 29 }%
\latex{ 40 }%
\latex{ 5 }%
Figure 14
Let us first calculate what percentage of all the votes got the candidates so far:
A: \latex{ 26 }% of the \latex{ 90 }% of the votes,
\latex{0.9\times0.26=0.234},
\latex{ 23.4 }% of all the votes;
B: \latex{ 29 }% of the \latex{ 90 }% of the votes,\latex{0.9\times0.29=0.261},
\latex{ 26.1 }% of all the votes;
C:\latex{ 40 }% of the \latex{ 90 }% of the votes,\latex{0.9\times0.40=0.360},
\latex{ 36 }% of all the votes.
Solution (a)
If the remaining \latex{ 10 }% are all for candidate \latex{ B }, then he can still win the election mathematically.
Solution (b)
Candidate \latex{ A } cannot win anyhow, because even if he gets all the \latex{ 10 }%, he still cannot catch up with candidate \latex{ C }.
Solution (c)
Let \latex{ x } denote the percentage in question. Candidate \latex{ C } will surely win, if
\latex{0.9\times\frac{x}{100}\gt0.1},
which implies
\latex{x\gt11.\dot{1}}.
So if candidate \latex{ C } has a lead of at least \latex{ 11.2 }% compared to the candidate after him with \latex{ 90 }% of the votes processed, then he will surely win.
Exercises
The citizens of planet Bog live in a democracy, therefore the citizens elect delegates to the assembly of the planet every fourth bog year. After the parliamentary election of \latex{ 9,572 } the data below appeared.
The number of eligible voters is: \latex{ 8,100,000 }. The ratios of the participants at the last four elections were:
\latex{ 9,560:53 }%, \latex{ 9,564:58 }%, \latex{ 9,568:61 }%, \latex{ 9,572:70 }%.
The ratio of the citizens who turned up for the election during the day (compared to the number of eligible voters) is shown in the diagram above (the election is on a specific day from \latex{ 6 } am till \latex{ 7 } pm).
\latex{ 110 } tons of paper was used for the election.
A \latex{ 160\, km } long string was created to tie up the \latex{ 11,000 } ballot boxes; the string was created using the colours of the flag of the planet. (The ballot boxes have the same size: \latex{50 \text{cm}\times50 \text{cm}\times 40 \text{cm}}.)
the ratio of the participants (%)
o'clock
\latex{ 19 }
\latex{ 17 }
\latex{ 15 }
\latex{ 13 }
\latex{ 11 }
\latex{ 9 }
\latex{ 10 }
\latex{ 20 }
\latex{ 30 }
\latex{ 40 }
\latex{ 50 }
\latex{ 60 }
\latex{ 70 }
\latex{ 70 }%
\latex{ 58 }%
\latex{ 46 }%
\latex{ 35 }%
\latex{ 28 }%
\latex{ 10 }%
The costs of the election (the money used on the planet is called dig):
\latex{ 2.5 } billion dig for the computer network with the help of which the votes are totalised \latex{ + 700 } million dig for the printing expenses \latex{ + 400 } million dig for the notes sent out \latex{ + } half billion dig for the wages.
The stake of the election is whether the blues or the greens will govern the planet. Who gets more than half of the valid votes is the winner. The pie chart shows the result of the voting.
green
\latex{49\%}
\latex{48\%}
\latex{3\%}
blue
invalid
{{exercise_number}}. Represent the ratio of the participants of the election on a column chart.
{{exercise_number}}. How many more citizens took part in the election in \latex{ 9,572 }than in \latex{ 9,568 }, if \latex{ 128,000 } more citizens were allowed to vote in \latex{ 9,568 } than in \latex{ 9,572 }?
{{exercise_number}}. The voting is valid if at least \latex{ 30 }% of the eligible voters go to give their votes. At what time do we know for sure that the voting is valid?
{{exercise_number}}. If a truck can transport \latex{ 8 } tons of paper and the truck is \latex{ 4.8\, m } long, then how many trucks are needed to transport the necessary paper? If these park in a queue after each other, then how long will the queue be?
{{exercise_number}}. Is the string created enough if the ballot boxes are tied up as it can be seen in the figure?
a)
b)
{{exercise_number}}. How much is the total cost of the elections in dig?
{{exercise_number}}. Represent the distribution of the costs on a pie chart.
{{exercise_number}}. How many more voters gave their votes for the blues than for the greens?
{{exercise_number}}. Ask more questions in connection with the data.