# Proof of the total probability rule for expected value?

• theone
In summary, the conversation discusses the concept of expected value and its relation to the probability of certain events occurring. The formula for calculating expected value in the presence of a scenario S is given as E(X) = E(X|S)P(S) + E(X|S_c)P(S_c), where X is a random variable and S is a scenario that affects the likelihood of X. The expected value is defined as the probability-weighted average of the possible outcomes of X. The conversation also mentions the need for a mathematical definition and properties of probability for proving the formula.
theone

## Homework Statement

Does anyone know of a simple proof for this: https://s30.postimg.org/tw9cjym9t/expect.png

E(X) = E(X|S)P(S) + E(X|S_c)P(S_c)

X is a random variable,
S is an a scenario that affects the likelihood of X. So P(S) is the probability of the scenario occurring and and P(S_c) is the probability of the scenario not occurring

## The Attempt at a Solution

How do you define ##E(X)##?

PeroK said:
How do you define ##E(X)##?

the expected value of the random variable X; the probability weighted average of the possible outcomes of X

theone said:
the expected value of the random variable X; the probability weighted average of the possible outcomes of X

You can't prove anything with just words. You need a mathematical definition.

PeroK said:
You can't prove anything with just words. You need a mathematical definition.

Do you mean this:
##E(X) = \sum_{i} X_i P(X_i)##

theone said:
Do you mean this:
##E(X) = \sum_{i} X_i P(X_i)##

If you have some properties for ##P## you could take it from there.

Not sure about a mathematical proof, but doesn't that formula just state the obvious? Perhaps putting it into words makes it clearer.

The expected value of X is the sum of the expected value of X when S happens multiplied by the probability that S happens plus the expected value of X when S doesn't happen times the probability of S not happening.

Because S happening and S not happening are mutually exclusive you can just add the two values together.

For a mathematical proof, you'd probably want to include your definition of E(X), the fact that P(S) + P(S') = 1, and the basic conditional probability formula (https://en.wikipedia.org/wiki/Conditional_probability)

Then go from there.

theone said:
Do you mean this:
##E(X) = \sum_{i} X_i P(X_i)##

You need formulas for ##E(X|S)## and ##E(X|S_c)##. Do you know what they are?

## What is the total probability rule for expected value?

The total probability rule for expected value is a mathematical formula that calculates the expected value of a random variable by multiplying the probability of each possible outcome by its corresponding value and summing the results.

## Why is the total probability rule important in statistics?

The total probability rule is important in statistics because it allows us to calculate the expected value of a random variable, which is a key concept in understanding probability distributions and making predictions based on data.

## Can you provide an example of how the total probability rule is used in real-life situations?

One example of how the total probability rule is used in real-life situations is in insurance. Insurance companies use the total probability rule to calculate the expected payout for various insurance policies based on the likelihood of different events occurring.

## How does the total probability rule differ from the law of total probability?

The total probability rule and the law of total probability are different concepts. The total probability rule is a formula used to calculate expected value, while the law of total probability states that the sum of the probabilities of all possible outcomes in a sample space is equal to 1.

## Are there any limitations to using the total probability rule for expected value?

One limitation of the total probability rule for expected value is that it assumes all outcomes are equally likely, which may not always be the case in real-world scenarios. It also assumes that the sample space is finite and exhaustive, which may not always be true.

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