Confusion about conditional probability

[TheMathNoob]

Homework Statement

Suppose that 70% of the statisticians are shy, whereas 30% of the economist are shy. Suppose also that 80% of the people at a large gathering are economists and the remaining 20% are statisticians. If you randomly meet a person at the gathering and the person is shy, what is the probability that the person is a statistician?
How can you interpret this statement?
Suppose that 70% of the statisticians are shy.

It sounds to me like the prob that we choose a person who is and statistician and is shy
but it is the prob that we choose a person who is shy given that he is an statistician

Homework Equations

The Attempt at a Solution

 

[andrewkirk]

Bayes formula provides the solution to this.

 

[TheMathNoob]

Bayes formula provides the solution to this.

this is my problem

How can you interpret this statement?
Suppose that 70% of the statisticians are shy.

It sounds to me like the prob that we choose a person who is and statistician and is shy
but it is the prob that we choose a person who is shy given that he is an statistician

 

[Ray Vickson]

Homework Statement

Suppose that 70% of the statisticians are shy, whereas 30% of the economist are shy. Suppose also that 80% of the people at a large gathering are economists and the remaining 20% are statisticians. If you randomly meet a person at the gathering and the person is shy, what is the probability that the person is a statistician?
How can you interpret this statement?
Suppose that 70% of the statisticians are shy.

It sounds to me like the prob that we choose a person who is and statistician and is shy
but it is the prob that we choose a person who is shy given that he is an statistician

Homework Equations

The Attempt at a Solution

You have it backwards. The person you meet is shy---that is, you are given that he/she is shy. So, you want P(stat|shy). Note that you already know the opposite conditional probability P(shy|stat), which was specified as part of the input data for the problem

 

[PeroK]

Homework Statement

Suppose that 70% of the statisticians are shy, whereas 30% of the economist are shy. Suppose also that 80% of the people at a large gathering are economists and the remaining 20% are statisticians. If you randomly meet a person at the gathering and the person is shy, what is the probability that the person is a statistician?
How can you interpret this statement?
Suppose that 70% of the statisticians are shy.

It sounds to me like the prob that we choose a person who is and statistician and is shy
but it is the prob that we choose a person who is shy given that he is an statistician

Homework Equations

The Attempt at a Solution

To understand these sort of questions I suggest drawing a probability tree and working backwards from right to left.

This allows you to see all the cases where there is the specified outcome (here that someone is shy) and allows you to map back to calculate the likely source of the outcome: Of all the shy people how many of them are statisticians.

 

[haruspex]

You have it backwards. The person you meet is shy---that is, you are given that he/she is shy. So, you want P(stat|shy). Note that you already know the opposite conditional probability P(shy|stat), which was specified as part of the input data for the problem

I think MathNoob is asking for clarification on one of the stated facts, not yet to the point of addressing the question.

 

[haruspex]

It sounds to me like the prob that we choose a person who is and statistician and is shy

No, that would match "70% of people are shy statisticians".
The "of" indicates the population to be sampled. 70% of statisticians have property P means that if we sample all the statisticians then 70% of them will have the property, so if we pick one at random then there is a 0.7 probability she will have that property.

 

[HallsofIvy]

Homework Statement

Suppose that 70% of the statisticians are shy, whereas 30% of the economist are shy. Suppose also that 80% of the people at a large gathering are economists and the remaining 20% are statisticians. If you randomly meet a person at the gathering and the person is shy, what is the probability that the person is a statistician?
How can you interpret this statement?
Suppose that 70% of the statisticians are shy.

It sounds to me like the prob that we choose a person who is and statistician and is shy
but it is the prob that we choose a person who is shy given that he is an statistician

Here's how I would do this: Imagine 100 people at the gathering. 80% of them, 80, are economists and 20%, 20, are statisticians. 70% of the statisticians, .70(20)= 14, are shy and 30% of the economists, .3(80)= 24, are shy. So there are a total of 14+ 24= 38 shy people, and 14 of them are statisticians. "Given that the person is shy" means that we restrict ourselves to only shy people and determine what percentage of them are statisticians.

Relevant equations

The Attempt at a Solution

[/QUOTE]