What Is the Difference Between Expected Value and Probability?

[V0ODO0CH1LD]

The expected value of a random variable is not necessarily the outcome you should expect. For discrete probability it might not even be a possible outcome for the experiment. So what does the expected value mean intuitively?

I will use and example because it helps me formulate my question:

Say you roll 3 dice; if you get at least one 6 you win, otherwise you loose. Now, the random variable is the number of 6's you get, so the expected value is 1/2. What does that mean? That if I play 10 times I would roll five 6's? In that case; if I play once, should I expect 1/2 of a 6? Doesn't that mean that I have a 50% probability of rolling a 6? But then I know that the probability of winning is less than 50%.

What am I thinking wrong?

 

[mathman]

Don't get hung up on the word expected. The expected value is just another way of saying average. For example one throw of one die has an expected value of 3.5, which is obviously something you will never get.

 

[V0ODO0CH1LD]

In that case, what does the average of a population mean intuitively? If it is not the most likely thing to be found in a population (guess that would be the mode, right?), then what is it? The central tendency? How can something be called the central tendency and at the same time not even be a possible outcome?

 

[chiro]

It's a mathematical attribute of a distribution and it's used for both practical and theoretical reasons.

When you want to estimate a population parameter using a sample, you construct an estimator and this has a mean and a variance. There are many results in statistics that say how many classes of estimators behave and what distributions they end up taking and knowing the mean and the variance helps you construct confidence intervals for hypothesis testing.

With regard to your question about 3.5, you need to consider in a discrete population whether you get something being between two values which implies that you will get the average swing between two specific values in your probability space.

Also there is a non-parametric measure called the median which always gives a specific value from the distribution (i.e. an event), but the median is more difficult to work with theoretically (although it may be required depending on how skewed the distribution is amongst other factors).

So to understand what expectation is used, you should consider how the expectation makes a lot of things easier in terms of theoretical results and calculations as opposed to using a mean.

The same sort of thing comes up when considering how to calculate variance: should we just use the absolute differences of values from the mean or the sum of squared differences?

Again when you look at the theory, it seems that using the sum of squares is a much better choice and this choices affects a tonne of statistical theory and all these different results link together when using the sum of squared differences.

If you are interested, try and find some articles on the differences between the mean and median and advantages and disadvantages for each in both a statistical and probabilistic sense.

 

[blue_raver22]

The expected value of a random variable is a mathematical concept that represents the average outcome of an experiment over a large number of trials. It is calculated by multiplying each possible outcome by its corresponding probability and summing them together. In your example, the expected value of rolling three dice and getting at least one 6 is indeed 1/2, which means that if you were to play this game many times, the average number of 6's you would roll would be 1/2. However, this does not mean that you can expect to roll exactly 1/2 of a 6 in a single trial or that you have a 50% chance of rolling a 6 in one roll.

The expected value is simply a theoretical concept that helps us understand the long-term behavior of a random variable. It does not guarantee that a particular outcome will occur in any given trial. In your example, the probability of winning the game is actually less than 50%, as you correctly pointed out. This is because the expected value takes into account all possible outcomes, including those that may not result in a win.

In essence, the expected value provides a way to quantify the likelihood of different outcomes in an experiment, but it should not be taken as a guarantee of a specific outcome in any given trial. It is important to understand the distinction between expected value and probability in order to accurately interpret and use these concepts in scientific research.