How do I find the odds for a voter in a problem involving different percentages?

Question

The problem is as follows:

In a certain community $30\%$ of the registered voters are members of the Blue party, $45\%$ of the registered voters belong to the Green party and the rest belong to the Orange party. At a recent election for choosing the new mayor, $20\%$ of Blue party supporters went to vote, $25\%$ of Green party supporters went to vote, and only $10\%$ supporters of Orange party went to vote. If a voter is selected at random What is the probability that he has voted?

What I tried to do to solve this problem was to account the total percentage by considering the contribution for which each supporter gives to the total based on the turnout.

Therefore:

$\textrm{Blue party turnout:}$

$\frac{20}{100}\times\frac{30}{100}=\frac{6}{100}$

$\textrm{Green party turnout:}$

$\frac{25}{100}\times\frac{45}{100}=\frac{45}{400}$

$\textrm{Orange party turnout:}$

$\frac{10}{100}\times\frac{25}{100}=\frac{10}{400}$

Then I assumed that the total percentage should be the odds or probability to find a voter who has casted its vote.

$\frac{6}{100}+\frac{45}{400}+\frac{10}{400}=\frac{79}{400}$

By expressing the later in scientific notation I end up with $19.75\times10^{-2}$ or $0.1975$ so I concluded that this should be the answer.

But is it okay to assume this? The way how I proceeded to calculate the odds is correct?. What would be the best way to solve this problem without incurring into errors of perception?. If what I did was correct is there a conceptual or mathematical justification? I'd like someone could guide me on this.

Answer 1

You did well.

More in probability terms:

Let $V$ denote the event that the selected person voted.

Let $B$ denote the event that the selected person belongs to Blue party.

Let $G$ denote the event that the selected person belongs to Green party.

Let $O$ denote the event that the selected person belongs to Orange party.

At first hand - because every person in the community will belong to exactly one of the $3$ mentioned parties - we have:

$$P(V)=P(V\cap B)+P(V\cap G)+P(V\cap O)$$

This can be rewritten as: $$P(V)=P(V\mid B)P(B)+P(V\mid G)P(G)+P(V\mid O)P(O)=$$$$\frac{20}{100}\frac{30}{100}+\frac{25}{100}\frac{45}{100}+\frac{10}{100}\frac{25}{100}=\frac{79}{400}$$

Answer 2

Imagine 100000 voters. 30% of them, 30000, are "Blue party", 45% of them, 45000, are "Green party", and 25% of them, 25000, are "Orange party". 20% of the "Blue party", .2(30000)= 6000, voted, 25% of the Green party, .25(4500)= 11250, voted, and 10% of the "Orange party", .10(25000)= 2500, voted.

That is a total of 6000+ 11250+ 2500= 19750 out of 100000 who voted.

The probability that a random person voted is 19750/100000= 0.19750.