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You have learned in prior chapters that a percentage may be converted to a decimal and multiplied by a number to find the percentage of that number. We will use this skill to find what is called the **expected value** of a variable, $x$. The expected value is the probability of an outcome. It tells us what the **most likely outcome** will be. Now, let’s make something very clear:

**The expected value is not the same as the mean value.**

The mean of a value tells us the average of all the numbers we have. The expected value tells us the most probable **outcome** of probabilities. The key idea to remember is that you will only ever use **expected value** with probabilities, and the problem statement will be very clear that you are tasked to find the expected value.

Now, let’s figure out how to find the expected value.

To find the expected value, we will calculate the sum of all the possibilities multiplied by their probability of happening. Let’s use an example to understand the formula for expected value.

Day of the week | Chance of rain | Predicted height of rainfall (inches) |

Monday | 20% | 2 |

Tuesday | 10% | 1 |

Wednesday | 0% | 0.5 |

Thursday | 35% | 0.5 |

Friday | 35% | 1 |

Using the table of data above and assuming that it will rain one day next week, what is the expected height of rainfall in inches?

To solve this problem, we will go across each row and multiply the chance of rain (the probability) by the predicted height of rainfall (the possible height of rainfall) and sum together all of the values. This concept is a little easier to understand if you say it out loud or in your mind, like this: “There’s a 20% chance that 2 inches of rain will fall on Monday, a 10% chance that 1 inch of rain will fall on Tuesday…”

$(0.2∗2)+(0.1∗1)+(0∗0.5)+(0.35∗0.5)+(0.35∗1)=1.025$

The expected height of rainfall on any given day is $1.025$ inches.

Hopefully, you can see what we did to solve the above problem. The formula for the expected value is given below. $E(X)$ is the expected value, $x$ is the possibility, $P$ is the probability, and the formula ends in “$…$” to show that it continues on for as long as you have data. Try to connect the formula below with what we did in the example above.

$E(X)=x_{1}∗P_{1}+x_{2}∗P_{2}+x_{3}∗P_{3}+…$

In our earlier example, $x_{1}$ was $2$ inches and our $P_{1}$ was $20%$. We multiplied across each row and found the sum of all of their products, just as the formula shows. Now, let’s do a simpler version of our first example to test our understanding of how to use the formula.

What is the expected value of $X$ given it has the following values and probabilities?

4 | 22% |

2 | 45% |

1 | 10% |

6 | 7% |

7.5 | 14% |

10 | 2% |

(spoiler)

Using the formula

$E(X)E(X)E(X) =x_{1}∗P_{1}+x_{2}∗P_{2}+x_{3}∗P_{3}+…=4∗0.22+2∗0.45+1∗0.1+6∗0.07+7.5∗0.14+10∗0.02=3.55 $

the expected value is $3.55$.

Another key feature of these problems is that the probabilities always add up to $100%$. This means that for any given situation, what we’re actually calculating is the value we can expect. This also means that the expected value should be somewhere between the lowest value and the highest value. If you get an answer that’s higher or lower than any other value in the table, redo your math because you may have typed in something wrong. Always double-check! Again, it’s a little hard to understand the concept until you have done a couple of problems, but you’ll get the hang of it!

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