# Probability of a Percent given a Percent

#### tizpan

##### New member
I'm hoping that I can gain some insight from fellow users in how to start off a probability question when given a percent 'given that' and another percent.

For example, if an election is split into percentages of votes per party and you know the party allegiance percentages of the town, how can you devise the probability that a random townsperson voted a certain way?

There are two different wholes at play: the voting population and the amount of voting percentages within the party. I can't figure out where to start

#### CaptainBlack

##### Well-known member
I'm hoping that I can gain some insight from fellow users in how to start off a probability question when given a percent 'given that' and another percent.

For example, if an election is split into percentages of votes per party and you know the party allegiance percentages of the town, how can you devise the probability that a random townsperson voted a certain way?

There are two different wholes at play: the voting population and the amount of voting percentages within the party. I can't figure out where to start
I doubt that you can, first you don't know the turn-out, or the turn-out for each party as these may be different.

Also the basic assumptions on voting behaviour do not seem reasonable.

CB

#### tizpan

##### New member
That is where I am getting confused in starting this problem.

The details given are as follows: A certain town is made up of 38.6% brown haired people, 57.1% blondes and 4.3% redheads. In the last town mayor race, votes were cast by 43.1% of the brown haired people, 40.7% blondes, and 51.7% of redheads. If a mayoral race voter is chosen at random, what is the probability that they are a brown haired?

Instinctively, I would look at this as the P(of brown haired in the town) * P(of brown haired that voted). So, .386*.431 equaling .166366. Somehow I can't help but think that there is more that I need to consider.

#### CaptainBlack

##### Well-known member
That is where I am getting confused in starting this problem.

The details given are as follows: A certain town is made up of 38.6% brown haired people, 57.1% blondes and 4.3% redheads. In the last town mayor race, votes were cast by 43.1% of the brown haired people, 40.7% blondes, and 51.7% of redheads. If a mayoral race voter is chosen at random, what is the probability that they are a brown haired?

Instinctively, I would look at this as the P(of brown haired in the town) * P(of brown haired that voted). So, .386*.431 equaling .166366. Somehow I can't help but think that there is more that I need to consider.
Bayes' theorem:

$P(Br|V)=P(V|Br)P(Br)/P(V)$

$$P(V|Br)=0.431$$, $$P(Br)=0.386$$,

\begin{aligned}P(V)&=P(V|Br)P(Br)+P(V|Bl)P(Bl)+P(V|R)P(R)\\&=0.431 \times 0.386+0.407 \times 0.571 + 0.517 \times 0.043 \approx 0.421 \end{aligned}

CB

#### tizpan

##### New member
That makes sense now, thank you Captain!