# Difference in probability and statistics?

1. May 4, 2005

### vptran84

Hello,

Just wondering what's the difference between probability and statistics? I can't seem to grasp the concept of it, can someone please explain the difference to me in Lehman's term? Also can someone give me an example of each? thanks in advance

Last edited: May 4, 2005
2. May 5, 2005

### honestrosewater

I think the distinction you want is that probability theory is pure math, while statistical theory is applied math. Statistics is the application of probability theory to the real world; It's a science, like physics, where you gather data, perform experiments, make predictions, and so on. So just as a physicist might use calculus to predict the path of a moving object, a statistician might use probability theory to predict the weather.

3. May 5, 2005

### kleinwolf

Let suppose you have a balanced coin....The probabilities are given before (a priori datas) as p(head)=1-p(tail)=.5

You throw the coin 5 times : you get : h,h,t,t,h

The statistics are : s[h]=3/5=.6, s[t]=1-s[h]=.4=2/5

The not-always-admitted assumption of prob. theory is that p(h)=limit s[h] when the number of trials for s[h] goes to infinity.

(Now of course you can make a MODEL, where the probabilities depend on the statistics....but this is no more axiomatic prob. theory, but just a toy model.)

Since the statistic is based on an experimental data, why should this limit exist for example ? In fact this question is not even possible, since no experiment allow to throw an infinite number of time.

But also consider a mathematical example : $$X(n)=\left\{\begin{array}{cc} h & \textrm{if the nth decimal of } \pi\textrm{ is n mod 10}\\ t & \textrm{ else }\end{array}\right.$$

Does the limit exist ?

4. May 5, 2005

### vptran84

i think i understand what you guys are saying...thank you for your responses.

5. Apr 4, 2011

### tonym144

Simply put, probability deals with what SHOULD occur, statistics deals with what HAS occurred. It's simply a matter of when.

6. Apr 4, 2011

### HallsofIvy

Staff Emeritus
In a certain sense, probability and statistics are opposites. Probability starts from a given probability distribution, with given parameters, and gives the chances that a specific outcome with happen. Statistics start with specific outcomes (the sample) and gives the parameters for the probability distribution.

7. Apr 18, 2011

### moonman239

Seconded.

As another rough example, let's say I roll a 6-sided dice 20 times.

Probability theory says that a 4 should happen only about 3 times (3 1/3 times, to be exact).

The statistics of the dice rolls say that a 4 actually happened 3 times (FYI, you will rarely if ever have a 4 showing up 3 times in 20 rolls.)

Statistical theory tells me that, at a 68.26895% confidence interval, the actual probability of a 4 is between 6.66666667% and 23.333333333%.